UC-NRLF 


m. 


REESE  LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

Deceived 
ccession  No.    7  / <£  )/ /   Class  No. 


THE 


CHARACTERS  OF  CRYSTALS 


AN  INTRODUCTION 


TO 


PHYSICAL  CRYSTALLOGRAPHY 


BY 


ALFRED  J.  MOSES,  E.M.,  PH.D. 

Professor  of  Mineralogy 
Columbia  Unvver$ity,  New  York  City 


NEW  YORK 

D.  VAN   NOSTRAND    COMPANY 
1*9* 


COPYRIGHT  1899 

BY 
ALFRED  J.  MOSES 


77.2-4 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY, 
LANCASTER,  PA. 


PREFACE. 

I  have  attempted,  in  this  book,  to  describe,  simply  and  con- 
cisely, the  methods  and  apparatus  used  in  studying  the  physical 
characters  of  crystals,  and  to  record  and  explain  the  observed  phe- 
nomena without  complex  mathematical  discussions.  In  the  last 
chapter  the  graduate  course  in  Physical  Crystallography  given  in 
Columbia  University  has  been  outlined. 

The  works  most  consulted  have  been  Grundriss  der  Physikalis- 
chcn  Krystallographie,  1 896,  by  T.  Liebisch  ;  Physikalische  Krys- 
tallographie ,  1895,  by  P.  Groth  ;  Traite  de  Crystallographie, 
1884,  by  E.  Mallard;  Treatise  oil  Crystallography,  1839,  by  W. 
H.  Miller;  Crystallography,  1895,  by  N.  Story-Maskelyne. 
Footnote  references  are  also  given  to  many  important  articles. 

It  is  hoped  the  book  will  be  found  useful  to  organic  chemists, 
geologists,  mineralogists  and  others  interested  in  the  study  of 
crystals. 

A.  J.   M. 

MlNERALOGICAL  DEPARTMENT, 
COLUMBIA  UNIVERSITY,  NEW  YORK  CITY, 
MARCH,  1899. 


CONTENTS. 

PART  I.   GEOMETRICAL  CHARACTERS. 

CHAPTER  I.   INTRODUCTORY 1-9 

CHAPTER   II.    THE    GENERAL    GEOMETRIC    PROPERTIES   OF    CRYS- 


ERRATA. 


In  Figs.  41,  89,  190  the  axis  of  composite  symmetry  should  be  dotted 
rig.  1 80  is  upside  down. 
P.  50  for  \h  h  2  h  i\  read  \h  h  2  hi}, 
P-  53>  line  7,  for  dehexagonal  read  dihexagonal. 
P-  56,  line  4,  for  Fig.  m  read  Fig.  162. 
P.  82,  line  20,  forOA=i.732readOAXi.732. 
P.  84,  line  5,  for  O  read  O'. 
P.  96,  last  line,  for  Weinschenk  read  Weinshenk. 
'114   Hne  next  to  last  for  ^  =  .009  read  A  =  530  /*/*. 
P.  123,  line  28,  for  two  circular  read  two  equal  circular 
P.  142,  line  5  from  the  bottom,  for  acute  read  obtuse 
P.  145,  lines  3  and  4,  for  146  and  147  read  118  and  119. 
P.  146,  lines  14  and  22,  for  146  read  118. 


dron,  47;  C.  19,  of  3°  ord.  Trigonal  Bipyramid,  4&]~U^2o^~oi  Jji- 
trigonal  Pyramid,  48;  C.  21,  of  Ditrigonal  Scalenohedron,  48;  C.  22, 
of  Ditrigonal  Bipyramid,  49;  C.  23,  of  3°  ord.  Hexagonal  Pyramid,  52  \ 
C.  24,  of  Hexagonal  Trapezohedron,  52  ;  C.  25,  of  3°  ord.  Hexagonal 
Bipyramid,  52;  C.  26,  of  Dihexagonal  Pyramid,  53;  C.  27,  of  Dihex- 
agonal Bipyramid,  53;  Projection  and  Calculation,  55.  ISOMETRIC:  C. 
28,  of  Tetartoid,  57;  C.  29,  of  Gyroid,  57;  C.  30,  of  Diploid,  58; 
C.  31,  of  Hextetrahedron,  58;  C.  32,  of  Hexoctahedron,  58;  Projec- 
tion and  Calculation,  61. 


OF  THE 

UNIVERSITY 


CONTENTS. 

PART  I.   GEOMETRICAL  CHARACTERS. 

CHAPTER  I.    INTRODUCTORY 1-9 

CHAPTER   II.    THE    GENERAL    GEOMETRIC    PROPERTIES   OF    CRYS- 
TALS    10-15 

Symmetry,  10;  Axes,  n;  Law  of  Rational  Indices,  n;  Param- 
eters, 12;  Indices,  12;  Determination  of  Elements  of  a  Crystal,  13. 

CHAPTER  III.    SPHERICAL  PROJECTION 16-24 

Zones,  1 6  ;  Symbol  for  Zone  Axis,  17  ;  Equation  for  Zone  Control, 
17  ;  Face  in  Two  Zones,  i  7  ;  Fourth  Face  in  a  Zone,  18  ;  Zone  of  Two 
Pinacoids,  19;  Zone  through  one  Pinacoid,  19;  Zones  in  which  two  In- 
dices are  Constant,  19;  Changing  Axes,  19;  Changing  Parameters,  20; 
Stereographic  Projection,  20;  Problems  in  Stereographic  Projection, 
21-24. 

CHAPTER  IV.  THE  THIRTY-TWO  CLASSES  OF  CRYSTALS.  .  .  25-62 
TRICLINIC:  C.  i,  Unsymmetrical,  26;  C.  2,  of  Pinacoids,  26; 
Projection  and  Calculation,  28.  MONOCLINIC  :  C.  3,  of  Sphenoid.  31; 
G.  4,  of  Dome,  31;  C.  5,  of  Prism,  31;  Projection  and  Calculation, 
33.  ORTHORHOMBIC  :  C.  6,  of  Bisphenoid,  35  ;  C.  7,  of  Pyramid,  35  ; 
C.  8,  of  Bipyramid,  36;  Projection  and  Calculation,  38.  TETRAGONAL: 
C.  9,  of  3°  ord.  Bisphenoid,  40;.  C.  10,  of  3°  ord.  Pyramid,  40; 
C.  n,  of  Scalenohedron,  41:  C.  12,  of  Trapezohedron,  41;  C. 
13,  of  3°  ord.  Bipyramid,  41  ;  C.  14,  of  Ditetragonal  Pyramid, 
42;  C.  15,  of  Ditetragonal  Bipyramid,  42;  Projection  and  Cal- 
culation, 45.  HEXAGONAL:  C.  16,  of  3°  ord.  Trigonal  Pyramid,  47; 
C.  17,  of  3°  ord.  Rhombohedron,  47;  C.  18,  of  Trigonal  Trapezohe- 
dron, 47;  C.  19,  of  3°  ord.  Trigonal  Bipyramid,  48;  C.  20,  of  Di- 
trigonal Pyramid,  48;  C.  21,  of  Ditrigonal  Scalenohedron,  48;  C.  22, 
of  Ditrigonal  Bipyramid,  49;  C.  23,  of  3°  ord.  Hexagonal  Pyramid,  52  ; 
C.  24,  of  Hexagonal  Trapezohedron,  52  ;  C.  25,  of  3°  ord.  Hexagonal 
Bipyramid,  52;  C.  26,  of  Dihexagonal  Pyramid,  53;  C.  27,  of  Dihex- 
agonal  Bipyramid,  53;  Projection  and  Calculation,  55.  ISOMETRIC:  C. 
28,  of  Tetartoid,  57;  C.  29,  of  Gyroid,  57;  C.  30,  of  Diploid,  58; 
C.  31,  of  Hextetrahedron,  58;  C.  32,  of  Hexoctahedron,  58;  Projec- 
tion and  Calculation,  61. 


vi  CONTENTS. 

CHAPTER  V.    MEASUREMENT  OF  CRYSTAL  ANGLES.    ...    .    .  63-75 

Application  Goniometers,  63 ;  Goniometers  with  Horizontal  Axes, 
64 ;  Goniometers  with  Vertical  Axes,  (56  ;  Errors  due  to  Imperfect  Cen- 
tering, 7 1  ;  Special  Cases  in  Measurement,  7  2 ;  Theodolite  or  Two- 
Circle  Goniometers,  74. 

CHAPTER  VI.  CRYSTAL  PROJECTION  OR  DRAWING 76-84 

Linear  Projections,  76;  Orthographic  Parallel  Perspective,  77; 
Clinographic  Parallel  Perspective,  79. 

PART  II.   THE  OPTICAL  CHARACTERS. 

CHAPTER  VII.  THE  OPTICALLY  ISOTROPIC  CRYSTALS 85-96 

Light  Rays,  85  ;  Ray  Surfaces,  85 ;  Ray  Front  and  Front  Normal, 
85.  CRYSTALS  WHICH  ARE  SINGLY  REFRACTING:  Ray  Surface,  86; 
Index  of  Refraction,  86 ;  Determination  by  Prism  Method,  88 ;  by 
Total  Reflection,  90.  CRYSTALS  WHICH  ARE  CIRCULARLY  POLARIZING, 
95.  Absorption,  96. 

CHAPTER  VIII.  THE  OPTICALLY  UNIAXIAL  CRYSTALS  .  .  .  .97-121 
CRYSTALS  IN  WHICH  THE  OPTIC  Axis  is  A  DIRECTION  OF  SINGLE  RE- 
FRACTION :  Double  Refraction,  97;  Plane  of  Vibration,  98;  Plane  of 
Polarization,  100  ;  Ray  Surface,  100;  Optical  Indicatrix,  101  ;  Deriva- 
tion of  Positive  Ray  Surface,  102  ;  Direct  Determination  of  Principal 
Indices  of  Refraction,  103.  INDIRECT  DETERMINATIONS  WITH  PLANE 
POLARIZED  LIGHT:  Interference,  106;  Poiariscopes,  106;  Phenomena 
with  Parallel  Monochromatic  Light  and  Crossed  Nicols,  no;  With 
Parallel  White  Light,  112;  Interference  Colors,  113;  Phenomena  with 
Convergent  Light  and  Crossed  Nicols,  115;  With  Parallel  Nicols,  116; 
Determination  of  Planes  of  Vibration  or  Extinction,  117  ;  of  Vibration 
Directions  of  Faster  and  Slower  Rays,  118  :  of  the  Retardation  A,  118; 
of  the  Strength  of  the  Double  Refraction,  119;  of  Thickness  of  Section, 
119;  Approximate  determination  of  Principal  Indices,  120;  Determina- 
tion of  Character  of  Ray  Surface  ,120. 

CHAPTER  IX.  THE  OPTICALLY  UNIAXIAL  CRYSTALS 

(Continued). 122-131 

CRYSTALS  IN  WHICH  THE  OPTIC  Axis  is  A  DIRECTION  OF  CIRCULAR 
POLARIZATION:  Plane,  Circular  and  Elliptical  Polarization,  122;  Rota- 
tion of  Plane  of  Polarization,  1^3;  Optic  Axis  a  Direction  of  Double  Re- 
fraction, 124;  Ray  Surface,  125;  Phenomena  with  Parallel  Monochro- 
matic Light  and  Crossed  Nicols,  126;  With  Parallel  White  Light,  127; 
With  Convergent  Light,  127  ;  Determination  of  Direction  of  Rotation, 
128;  of  Angle  of  Rotation,  129;  Absorption  and  Pleochroism,  130. 


CONTENTS.  vii 

CHAPTER  X.  THE  OPTICALLY  BIAXIAL  CRYSTALS 132-144 

Optical  Indicatrix,  133;  Ray  Surface,  133;  Positive  and  Negative 
Ray  Surfaces,  136  ;  Refraction,  136  ;  Phenomena  with  Parallel  Monochro- 
matic Light  and  Crossed  Nicols,  137  ;  With  Parallel  White  Light,  137  ; 
With  Convergent  Monochromatic  Light,  138;  With  Convergent  White 
Light,  140;  Distinctions  between  Orthorhombic,  Monoclinic  and  Tri- 
clinic  Crystals,  140. 

CHAPTER  XI.  DETERMINATION  OF  THE  OPTICAL  CHARACTERS  OF  BI- 
AXIAL CRYSTALS 145-162 

Orientation  and  Determination  of  the  Principal  Vibration  Directions, 
145 ;  Measurement  of  the  Principal  Indices,  146  ;  Determination  of 
Angle  between  Optic  Axes,  148  ;  of  True  Axial  Angle,  153  ;  Calculation 
of  Axial  Angle  from  Indices,  153  ;  Determination  of  Character  of  Ray  Sur- 
face, 154;  Crystals  in  Thin  Rock  Sections,  154;  Absorption  and  Pleo- 
chroism,  155. 

Absorption  Tufts,  158;  Metallic  Refraction  or  Metallic  Lustre, 
159;  Surface  Colors,  159;  Fluorescence,  160;  Phosphorescence,  161 ; 
Norremberg  and  Reusch  Combinations  of  Mica  Plates,  161. 

PART   III.   THE   THERMAL,    MAGNETIC   AND  ELEC- 
TRICAL CHARACTERS,  AND  THE  CHARAC- 
TERS DEPENDENT  UPON  ELASTIC- 
ITY AND  COHESION. 

CHAPTER  XII.  THE  THERMAL  CHARACTERS 163-171 

Heat  Conductivity,  164;  Expansion  by  Heat,  166;  Direct  Meas- 
urement of  Linear  Expansion,  167;  Measurement  of  Expansion  by 
Change  of  Diedral  Angles,  168;  Determination  of  Expansion  by 
Changes  in  the  Optical  Character,  169. 

CHAPTER  XIII.     THE  MAGNETIC  AND  ELECTRICAL  CHARACTERS  OF 

CRYSTALS 172-184 

The  Magnetic  Induction  of  Crystals,  172  ;  Strength  of  Magnetization 
in  Different  Directions  in  a  Crystal,  173;  Transmission  of  Electric  Rays, 
175;  Electrical  Conductivity,  175;  Thermoelectric  Currents,  176; 
Dielectric  Induction  in  Crystals,  177;  Pyro-Electricity,  180;  Piezo- 
Electricity,  182  ;  Theory  of  Pyro-  and  Piezo-Electricity,  183. 

CHAPTER  XIV.  ELASTIC  AND  PERMANENT  DEFORMATION  OF  CRYS- 
TALS    ••'...  185-198 

Homogeneous  Elastic  Deformation,  185  \  Elastic  Deformation  Due 
to  Pressure  in  One  Direction,  185  ;  Surface  of  Extension  Coefficients, 


viii  CONTENTS. 

1 86;  Effect  of  Pressure  in  One  Direction  Upon  the  Optical  Characters, 
187  ;  Cleavage,  188  ;  Gliding  Planes,  190  ;  Parting,  igi  ;  Percussion  Fig- 
ures, 191;  Etch  Figures,  192;  Corrosion  Faces,  196;  Hardness,  196; 
The  Methods  of  Static  Pressure,  198. 

APPENDIX. 

SUGGESTED   OUTLINE  OF  A   COURSE   IN    PHYSICAL   CRYSTAL- 
LOGRAPHY   199-206 

Preliminary   Experiments,    199 ;    Systematic   Examination    of   the 
Crystals  of  Any  Substance,  203. 


PART  I.     GEOMETRICAL  CHARACTERS. 


CHAPTER  I. 


INTRODUCTORY. 


It  s  a  general  property  of  definite  chemical  substances  to  as- 
sume at  solidification  regular  forms  bounded  by  planes  and  obser- 
vation has  proved  that  the  forms  which  occur  are  related  to  each 
other  and  characteristic  of  the  substance. 

The  geometric  study  of  crystals  has  for  its  purpose  the  group- 
ing of  crystals  into  series,  each  of  which  shall  consist  of  the  forms 
in  which  one  substance  can  appear ;  the  determination  of  the  angles 
between  the  faces  of  crystals,  and  from  these  the  elements  and  sym- 


FlG.    I. 


FIG.  2. 


bols  of  the  faces,  or  conversely  the  determination  of  the  form  and 
angles  from  the  elements  and  the  face  symbols. 

The  term  crystal  originally  meant  the  angular  forms  of  the  sub- 
stance called  rock  crystal,  Figs.  I  and  2,  which,  according  to 
PLINY,*  was  only  "water  frozen  by  the  most  excessive  cold  and 
found  only  in  places  where  snow  is  changed  into  ice."  The  an- 

*  Quoted  by  Rome  Delisle,  Essai  de  Crystallographie  (1772),  p.  3. 


2  CHARACTERS  OF  CRYSTALS. 

gular  shapes  of  this  substance  and  of  garnet,*  beryl  and  possibly 
diamond,  were  known  to  the  ancients,  but  were  regarded  as  acci- 
dents and  no  general  property  was  suspected. 

In  1568  WENTZEL  jAMiTZER,f  a  Nuremburg  .goldsmith,  devel- 
oped in  perspective  over  one  hundred  and  forty  simple  and 
complex  shapes  from  the  geometric  tetrahedron,  octahedron, 
cube,  dodecahedron  and  icosahedron  by  replacing  all  similar  edges 
and  angles  by  one. or  more  planes,  and  the  famous  astronomer 
KEPLER,  in  1619,  developed  a  similar  series  of  figures.  Many  of 
these  shapes  correspond  to  crystals,  and  the  method  of  modifica- 
tion may  have  suggested  to  Rome  Delisle  the  method  used  by 
him  more  than  a  century  later. 

The  development  of  chemistry  from  alchemy  was  accompanied 
by  the  study  of  many  salts,  the  solutions  of  which,  on  evaporation, 
yielded  regular  and  constant  shapes  which  were  suspected  to  be  to 


FIG.  3. 

some  extent  at  least  characteristic  of  the  salt,  for  LIBAVIUS,^:  in 
1597,  stated  that  the  nature  of  the  saline  components  of  a  mineral 
water  could  be  ascertained  by  an  examination  of  the  crystalline 
form  of  the  salts  left  on  the  evaporation  of  the  water. 

In  1669  NICOLAS  STENO,§  a  Danish  anatomist,  announced  that  if 

*  "  There  is  also  an  incombustible  stone  found  about  Miletum  which  is  of  an  angu- 
lar shape,  and  sometimes  regularly  hexangular ;  they  call  this  also  a  carbuncle."  The- 
ophrastus's  History  of  Stones.  Trans,  by  Sir  John  Hill,  p.  77. 

•f-Perspectiva  Corporum  Regularum,  Quoted  in  Marx's  Geschichte  der  Krystalkunde. 

^Roscoe  £  Schorlemmer's  Treatise  on  Chemistry,  I.  705. 

$De  solido  intra  solidum  naturaliter  contento  dissertationis  prodromus.  Florentiae, 
1669:  English  translation,  London,  1671. 


GEOMETRICAL  CHARACTERS. 


different  specimens  of  rock  crystal  were  examined,  it  would  be 
found  that  in  spite  of  the  variation  in  the  relative  size  of  the  faces 
and  in  the  shape  of  the  crystal  there  was  no  variation  in  the  angles 
between  the  faces.  This  he  illustrated  by  figures  of  sections  at  right- 
angles  to  a  prism  edge  and  others  of  sections  at  right  angles  to  an 
edge  between  a  prismatic  and  a  pyramidal  face,  as  shown  in  Fig.  3. 
A  statement  showing  a  very  considerable,  advance  from  this 
was  made  by  DOMINICO  GULIELMINI  in  1704,  who  asserted  that 
every  salt  had  its  peculiar  shape  which  .never  changed .--and  that 
even  in  imperfect  and  broken  crystals  the  angles  were  constant. 

T.ORBERN  BERGMAN*  records  that  his  pupil  GAHN  having  broken 
a  piece  of  Dogtooth  Spar  observed  that  it  divided  into  little  rhom- 
bohedra.  In  studying  this,  Bergman  found  that  by  laying  together 
these  rhombohedra  in  certain  ways  he  could  build  up  either  a  hexa- 
gonal prism  terminated  by  this  form  or  the  common  scalenohedron 
of  calcite,  or  a  form  like  the  rhombic  dodecahedron,  all  of  which 
are  shown  in  Fig.  4. 

From  this  he  deduced  that  there  was  a  relation  be- 
tween outer  form  and  inner  structure  and  that  the  con- 
stituent parts  of  all  crystals  could  be  referred  to  a  very 
small  number  of  primitive  forms  which  could  be  found 
by  breaking  the  crystals. 

In  the  preparation  of  models  of  crystal  forms  ROME 
DELISLE,  about  1783,  measuredf  the  interfacial  angles 
directly  with  an  application  goniometer  devised  by 
Carangeot  for  this  purpose.     He  described  over  four 
hundred  crystal  forms  and  formulated  the  now  univer- 
sally accepted  law  of  constancy  of  interfacial   angles 
FIG.  4.        as  f0uows  :j  4</7/  spfte  Of  tjie  numberless  variations  of 
which  the  primitive  form  of  a  salt  or  a  crystal  is  capable,  one  thing 
never  vaties,  but  is  always  constant  in  each  species,  namely :   The  angle 
of  incidence  or  the  respective  inclination  of  the  faces  to  each  other." 

The  law  of  symmetry  may  also  be  said  to  be  based  on  a  law  of 
Delisle.  "  Every  face  has  a  similar  face  parallel  to  it." 

*  De  Formis  Crystallorum  cited  in  von  Kobell's  Geschichte  der  Mineralogie. — 
p.  81. 

\  The  few  measurements  of  crystals  which  had  thus  far  been  made  were  measure- 
ments of  plane  angles  only,  which  varied  with  differences  in  development  so  that  the 
interfacial  angles  calculated  from  them  were  often  incorrect. 

\  Cristallographie,  ou  description  der  formes  propres  &  tous  les  corps  du  regne- 
mineral.  Paris,  1783.  p.  70. 


CHARACTERS  OF  CRYSTALS. 


Delisle  also  developed  Bergman's  idea  of  primitive  form  and  by 
a  method  similar  to  that  of  Jamitzer  derived  a  series  of  secondary 
forms  from  each  primitive  by  replacing  the  edges  or  angles  of  the 
primitive  form  by  single  planes  or  groups  of  planes  in  such  a  way 
that  equivalent  geometric  parts  were  similarly  treated.  He  re- 
ferred all  forms  to  six  classes  of  primitive  form  :  The  regular  tet- 
lahedon,  Fig.  5,  and  the  cube,  Fig.  6,  which  permit  of  no  varieties, 
and  the  rectangular  octahedron,  Figs.  7,  8,  9  ;  the  rhomboidal  paral- 
lelopipedon  Figs.  10,  II,  12  ;  the  rhomboidal  octahedron  Figs.  13,  14, 
15,  and  the  dodecahedron  with  triangular  faces,  Fig.  1 6,  all  of  which 
were  susceptible  of  many  varietions  according  to  their  angles. 


13 


15 


16 


As  an  example  of  the  method  of  derivation  there  could  be  de- 
rived from  the  cube,  Fig.  17,  Fig.  18  by  truncating  each  edge, 
Fig.  19  by  bevelling  each  edge,  Fig.  20  by  truncating  each  solid 
angle,  Fig.  21  by  replacing  each  solid  angle  by  three  planes,  each 
cutting  equal  lengths  from  two  edges  and  a  different  length  from 
the  third,  Fig.  22  by  replacing  each  solid  angle  by  six  planes,  each 


GEOMETRICAL  CHARACTERS. 


5 


cutting  three  unequal  distances  on  the  three  edges,  and  many 
others.  Similarly  all  the  secondary  forms  of  any  compound  could 
be  derived  from  a  primitive  form,  and  thus  connecting  them  in  a 
definite  series.  Any  series  could,  however,  be  derived  from  almost 
any  member  of  the  series  in  the  manner  described  and  the  primi- 
tive forms  could  only  be  chosen  arbitrarily. 


18 


20 


The  establishment  of  crystallography  upon  a  mathematical  basis 
is  chiefly  due  to  the  ABBE  HAOY,  who,  apparently  in  ignorance  of 
the  discoveries  of  Bergman,  had  his  attention  directed  to  the  inter- 
nal structure  of  crystals  by  the  accidental  dropping  of  a  six-sided 
prism  of  calcite  which  broke  into  rhombohedral  fragments.  He 
found  that  this  property  of  "  cleavage  "  was  a  general  one  and  that* 
the  crystals  were  built  up  of  molecules  of  the  shape  of  the  cleavage 
forms.  Although  in  many  instances  no  cleavage  was  found  or 
cleavage  only  in  one  direction,  which  yielded  no  solid,  he  assumed 
that  such  solids  did  exist  in  all  crystals  and  were  the  true  primi- 
tive forms  upon  which  the  other  secondary  forms  depended  and 
that  the  shape  in  such  cases  could  be  determined  by  striations 
and  other  markings  on  the  faces  or  by  analogy  between  the  shapes 
of  the  secondary  forms  and  similar  forms  of  other  crystals  which 
did  show  cleavage. 

Like  Delisle  he  made  six  groups  of  primitive  forms,  of  which 
only  two  were  identical  with  those  of  Delisle  namely  the  tetrahe- 
dron, Fig.  5,  and  the  dodecahedron  with  triangular  faces,  Fig.  1 6. 

The  forms,  i.  PARALLEOPIPEDON.    Including   cube;  right  prisms 

*  Essai  d'une  theorie  sur  la  structure  des  cristaux  1784.  Journal  de  Physique 
XLIII,  p.  103. 

Also  in  Gren's  Neues  Journal  der  Physik,  1795,  VII.,  p.  418. 


6  CHARACTERS  OF  CRYSTALS. 

with  bases  square,  rectangular,  rhombic,  oblique ;  oblique  prisms 
with  bases  rhombic,  Fig.  28;  rectangular,  oblique;  rhomboktdron 
obtuse;  acute.  2,  REGULAR  TETRAHEDRON.  3.  OCTAHEDRON  WITH 
TRIANGULAR  FACES.  Including  the  regular  Octahedron  and  the 
octahedra  with  bases  rectangular,  square,  rhombic.  4.  SIX-SIDED 
PRISM.  5.  DODECAHEDRON  WITH  RHOMBIC  FACES.  6.  DODECAHE- 
DRON WITH  < TRIANGULAR  FACES. 

The  existence  of  cleavages  other  than  those  producing  the  prim- 
itive form  led  Haiiy  to  assume  the  existence  of  still  simpler  shapes 
which  he  called  integrant  molecules.  These  he  limited  to  three 
kinds.  The  tetrahedron,  the  trigonal  prism  and  the  parallelopip- 
edon. 

With  crystals  of  any  substance  Haiiy  discovered  that  all  forms 
other  than  the  primitive  form  could  be  exactly  imitated  by  build- 
ing on  the  faces  of  the  primitive  form  successive  plates  or  layers 
of  integrant  molecules  each  successive  layer  regularly  diminishing 
by  the  abstraction  of  one  or  more  rows,  either  parallel  to  each 
edge,  or  to  the  diagonals  of  the  faces  of  the  primitive  form,  or 
parallel  to  some  intermediate  line.  A  rhombic  dodecahedron 
might,  for  instance,  be  found  to  cleave  with  equal  ease  in  three 
directions  at  right  angles  to  each  other.  The  integrant  molecules 
would  then,  according  to  Haiiy,  be  cubes  and  the  kernel  or  prim- 


FIG.  23.  FIG.  24. 

itive  form  also  a  cube,  Fig.  23  ;  then,  as  perfect  cleavage  requires 
that  the  minute  integrant  molecules  be  so  piled  as  not  to  break 
joints,  the  structure  would  be  similar  to  that  shown  in  Fig.  24,  in 
which  the  successive  layers  piled  on  top  of  any  face  of  the  cubic 
primitive  form  regularly  diminish  one  row  at  a  time  on  each  edge. 


GEOMETRICAL  CHARACTERS.  7 

For  only  with  such  a  rate  of  decretion  can  the  pyramidal  planes 
sOI  and  tOI  unite  to  one  plane  sOIt  and  the  diedral  angle  between 
alternate  planes  be  ninety  degrees.  As  the  number  of  plates  added 
would  be  very  large  and  the  little  cubic  molecules  too  small  to  be 
separately  visible  the  steps  would  appear  to  lie  in  the  planes.' 

Another  example  may  be  given  to  show  that  this  method  of 
building  yields  forms  corresponding  to  actual  crystals.  Fig.  25 
shows  a  form  observed  on  cobaltite,  in  which  the  diedral  angle 
at  the  edge,/*?,  is  126°  52'.  Assuming  a  cubic  kernel,  Fig.  26.  Fig. 
27  shows  the  structure  according  to  Haliy,  in  which  the  decretions 


FIG.  25.  FIG.  26.  FIG.  27. 

correspond  to  a  triangle  with  a  base  of  2  and  a  perpendicular  of  7, 
that  is,  the  angle  at  the  base  has  a  tangent  equal  to  0.5,  or  is  26° 
<34r.  The  angle  at  p  q  is  twice  the  complement  of  this  ;  that  is, 
126°  52'. 

The  decretion*  was  symmetrical  that  is  it  was  repeated  on  all 
similar  parts  of  the  kernel  and  Hauy's  experiments  showed  that  the 
secondary  planes  usually  resulted  from  the  subtraction  of  one  or  of 
two  rows,  and  were  always  according  to  some  simple  rational  num- 
ber never  to  his  knowledge  exceeding  four.  This  is  essentially  the 
basis  of  the  law  of  rational  indices,  the  fundamental  law  of  crystal- 
lography. 

PROF.  BERNHARDI,  of  Erfurt,  pointed  out|  that  the  primitive 
form  should  be  chosen,  not  as  with  Haliy,  from  molecules 
of  nature,  but  according  to  convenience  and  fitness,  and  that 

*The  law  of  symmetry  was  stated  by  Haiiy  as  follows:  "It  consists  in  this,  that 
any  one  method  of  decretion  is  repeated  on  all  those  parts  of  the  nucleus  of  which  the 
resemblance  is  such,  that  one  can  be  substituted  for  the  other  by  changing  the  position 
of  this  nucleus  with  respect  to  the  eye,  without  it  (the  nucleus)  ceasing  to  be  pre- 
sented in  the  same  aspect."  Memoire  sur  une  loi  de  Cristallisation,  1815. 

f  Von  Kobell's  Geschichte  der  Mineralogie,  p,  197. 


8  CHARACTERS  OF  CRYSTALS. 

the  paralellopipedon  was  not  a  satisfactory  form  for  calculation 
in  many  cases,  as  it  was  often  an  open  form  the  height  of  which 
could  only  be  determined  from  some  secondary  face,  and  he  recom- 
mended the  choice  of  closed  primitive  forms  only  and  suggested 
seven,  six  of  which  are  still  regarded  as  the  simplest  forms  in  the 
crystal  systems,  namely :  Cube,  Rhombohedron,  Square  Octahedron, 
Rlwmbic  Octahedron,  Rhomboidal  Octahedron,  Triple  Rhomboidal 
Octahedron,  the  seventh  form  was  a  rectangular  octahedron  which 
can  be  referred  to  the  rhombic  octaJiedron. 

PROF.  WEISS,  of  Berlin,  devised  a  purely  geometric  mode  of  treat- 
ment in  which  he  discarded  entirely  the  idea  of  primitive  form,  say- 
ying  *  "  that  Haiiy's  hypotheses  of  decreted  rows  entangled  the 
problem  with  self-created  difficulties  "  and  that  the  "  mechanical- 
atomic  presentation  which  Hauy  deduced  should  be  stripped  away 
so  that  the  acquired  knowledge  of  the  mathematical  relation  should 
be  more  clearly  seen."  Weiss  assumed  the  existence  of  certain  fun- 
damental lines  or  axes  passing  through  a  common  center  and 
made  four  groups:  1st,  with  three  equal  axes  at  right  angles  to 
each  other;  2d,  with  three  axes  at  right  angles  to  each  other,  of 
which  two  were  equal;  3d,  with  three  axes  at  right  angles, and  all  dif- 
ferent lengths ;  4th,  with  three  equal  axes  at  sixty  degrees  to  each 
other  in  one  plane  and  one  at  right  angles  to  these,  but  of  different 
length. 

He  deduced  all  the  primitive  forms  of  Hauy  by  constructing 
planes  which  passed  :  First — Through  ends  of  three  lines.  Second 
— Through  ends  of  two  of  the  lines  and  parallel  to  a  third.  Third 
— Through  an  end  of  one  of  the  lines  and  parallel  to  two  of  them. 
By  taking  points  along  each  of  these  lines  at  twice,  three  times 
and  four  times,  etc.,  the  original  length,  and  constructing  planes 
in  the  same  way  as  before,  he  obtained  all  the  secondary  forms. 

The  lengths  of  the  semi-axes  of  the  primitive  forms  were  called 
a,  b  and  c,  and  the  position  of  any  face  was  denoted  by  the  ratio 
of  the  intercepts  in  terms  of  a,  b  and  c  ;  thus,  2a :  b :  ^c  or  a :  ^b  :  2c, 
and  so  on.  This  method  is  still  the  simplest  and  most  satisfactory 
if  only  an  elementary  knowledge  of  type  forms  is  desired,  but  is 
cumbersome  and  tedious  in  calculation. 

HAUSMANN  applied  spherical  trigonometry  to  crystallographic 
calculations  in  1803  and  BERNHARDif  in  1808  pointed  out  that  it 

*  Uebersichtliche  Darstellung    der  verschiedene  Abtheilung  der   Krystallisations- 
Systeme.     Denkschrift  der  Berliner  Akad.  der  Wissen.     1814-15.     p.  298. 
\  Gehlen's  Journal,  1808,  2,  378. 


GEOMETRICAL  CHARACTERS.  9 

would  be  better  to  determine  trigonometrically  the  relations  be- 
tween the  lines  from  a  common  point,  normal  to  each  face,  as  these 
were  the  directions  of  attraction  and  growth. 

PROF.  MOHS,  of  Freiberg,  grouped*  all  forms  in  four  SYSTEMS,  the 
Cubic,  Pyramidal,  Rhombohedral  and  Prismatic,  each  consisting  of 
a  series  of  forms  geometrically  derived  from  a  "  fundamental  form  " 
by  modifying  planes  which  cut  distances  from  certain  lines  in  the 
ratio  of  whole  numbers.  The  fundamental  forms  were  the  simplest 
closed  forms  and  were  not  selected  from  cleavage  or  structure,  but 
purely  for  geometric  simplicity.  In  1822  he  proved  the  existence 
of  two  more  Systems,  the  Monoclinic  and  Triclinic,  the  forms  of 
which  had  previously  been  considered  as  partial  forms  of  the  other 
four  systems. 

F.  C.  NEUMANN  in  1823  proposed  a  graphic  method  in  which 
the  crystal  faces  were  indicated  by  the  points  in  which  radii  drawn 
normal  to  the  faces  met  the  surface  of  a  circumscribing  sphere. 

In  1825  WHEWELLf  showed  that  if  a  solid  angle  of  the  primitive 
form  was  taken  as  the  origin,  and  the  edges  as  cordinates,  with 
lengths  x,  y,  z,  then  any  secondary  face  would  be  expressed  by  the 

A£         Al  & 

equation  7- *y.  T  =  m,  in  which  m  was  not  dependent  upon  the  in- 

rl     K    / 

clination  of  the  face,  and, therefore,  the  quantities  \  T'  T'  j  [      or 

say  { /,  q,  r  }  could  represent  the  face.  These  are  essentially 
the  reciprocals  of  the  Weiss  intercepts,  and  are  the  indices  used  by 
Miller. 

By  far  the  most  important  advance  since  Haiiy  was  the  method 
developed  J  by  PROFESSOR  W.  H.  MILLER,  of  Cambridge  Univer- 
sity, in  which,  by  spherical  trigonometry,  a  series  of  simple  sym- 
metrical expressions  were  deduced  for  the  normal  angles  in  terms 
of  the  reciprocal  intercepts  of  Whewell,  and  the  positions  of  faces 
were  indicated  by  stereographic  projection  of  the  points  in  which 
radii  perpendicular  to  the  faces  meet  the  surface  of  the  sphere. 
The  method  is  still  the  best,  and  is  gradually  supplanting  all 
others  except  for  elementary  work. 

*  Treatise  on  Mineralogy,  or  the  Natural  History  of  the  Mineral  Kingdom.  (Haid- 
inger's  translation),  Edinburgh,  1825. 

f  A  general  method  of  calculating  the  angles  made  by  any  planes  of  crystals.  Rev. 
W.  Whewell,  Transactions  Royal  Society,  1825,  CXV.,  87-180. 

\  A  treatise  on  Crystallography,    1839. 


CHAPTER  II. 


THE  GENERAL  GEOMETRIC  PROPERTIES  OF 
CRYSTALS. 


A  PLANE  OF  SYMMETRY  is  a  plane  through  the  centre  which 
divides  the  crystal  so  that  either  half  is  the  mirrored  reflection  of 
the  other,  and  every  line  perpendicular  to  the  plane  (and  within 
the  solid)  connects  corresponding  points  of  the  solid  and  is  bisected 
by  the  plane.  Every  plane  of  symmetry  is  necessarily  parallel  to 
a  possible  crystal  face,  for  if  two  faces  are  symmetrical  to  it,  it 
must  pass  through  their  edge,  that  is  to  be  in  the  same  zone.  So, 
also,  with  a  second  pair  of  faces.  But  any  plane  lying  in  two 
known  zones  has  rational  indices  (p.  11)  and  is  a  possible  crystal 
face,  as  will  be  shown  later. 

One  plane  of  symmetry  may  occur  alone.  If  two  planes  occur 
they  must  be  at  right  angles,  and  they  make  necessary  a  third  at 
right  angles  to  both. 

AN  Axis  OF  SYMMETRY  is  a  line  through  the  centre  such  that  a 
revolution  of  180°  or  less  around  it  will  bring  a  face  into  coinci- 
dence with  the  original  position  of  an  equivalent  face. 

The  intersection  of  two  planes  of  symmetry  is  necessarily  an 
axis  of  symmetry.  Every  axis  of  symmetry  is  either  parallel  to 
an  edge  between  two  possible  faces  or  normal  to  a  possible  face. 
It  has  been  proved*  that  that  there  can  be  only  four  kinds  of  axes 
of  symmetry :  Binary,  Ternary,  Quaternary  and  Senary  in  which 
equivalent  faces  become  coincident  by  revolutions  of  180°,  120°, 
90°  and  60°  respectively.  These  may  occur  alone  or  in  combi- 
nations or  with  planes  of  symmetry. 

There  may  be  distinguished  three  methods  by  which  equivalent 
planes  equally  distant  from  the  centre  may  be  made  to  coincide : 

1st.  By  reflection  in  a  plane  of  symmetry. 

2d.  By  rotation  around  an  axis  of  symmetry. 

*  Groth's  Physikalische  Krystallographie,  p.  313,  III.  edition. 


GEOMETRICAL  CHARACTERS.  n 

3d.  By  composite  symmetry  or  simultaneous  rotation  around  an 
axis  and  reflection  in  a  plane. 

Composite  symmetry  can  occur  with  a  binary  axis,  and  plane 
perpendicular  thereto  ;  it  is  impossible  with  a  ternary  axis,  and  with 
quarternary  or  senary  axes  is  equivalent  to  simple  symmetry  with 
binary  or  ternary  axes  respectively. 

AXES.  The  faces  of  crystals  are  defined  in  position  by  referring 
them  to  imaginary  lines  called  axes.*  The  selection  is  largely  ar- 
bitrary, but  with  the  object  of  securing  the  simplest  indices.  For 
this  purpose  they  must  be  lines  parallel  to  the  edges  between  pos- 
sible planes  and  lines  as  important  to  the  symmetry  as  possible. 
Generally  three  directions  are  selected  which  are  parallel  to  the  in- 
tersections of  three  occurring  or  possible  faces. 

Let  XX  Y^T  ZZ,  Fig.  28,  be  three  such 
axes  and  let  all  distances  be  measured  on 
these  lines  from  their  origin  or  common  in- 
tersection O,  distances  in  the  direction  OX, 
OY  or  OZ  being  regarded  as  positive  and 
those  in  the  directions  OX,  OY  or  OZ 
as  negative. 

Any  plane  as  A  B  C  would  be  determined 
in  space  by  its  intercepts  OA,  OB,  OC,  on 
*  these  axes,  or  in  angular  position  by  the 

ratios  of  these  intercepts,  and  as  in  crystals 

there  is  constancy  in  inclinations  of  planes  to  each  other  and  there- 
fore to  the  axes,  but  no  constancy  in  the  linear  dimensions,  some 
expression  for  the  relative  instead  of  the  absolute  values  of  the  in- 
tercepts is  used  as  a  symbol,  which  may  be  the  intercepts  expressed 
as  a  proportion  or  some  condensed  expression  derived  from  the 
ratios  of  the  intercepts. 

The  Fundamental  Law  or  Laiv  of  Rational  Indices. 

Experience  teaches  that  a  simple  relation  exists  between  the  in- 
tercepts of  different  planes  of  the  crystals  of  any  chemical  sub- 
stance, and  that  if  the  axial  intercepts  of  any  face  be  divided  by  the 
corresponding  intercepts  of  any  other  face  the  quotients  will  be  only  sim- 
ple rational  numbers. 


*  Equivalent  axes  are  necessarily  surrounded  by  the  same  number  of  faces  placed 
in  the  same  way. 


12  CHARACTERS  OF  CRYSTALS. 

For  instance,  if  the  intercepts  of  some  face  ABC  are 
OA:  OB:OC  =  a:  b\ct 

then  these  values  divided  by  the  intercepts  of  any  other  face  H  K  L 
will  yield  quotients 

_^     _b_       *"./.*./ 
OH  :OK*  OL~ 

in  which  h,  k  and  /  are  simple  rational  numbers  such  as  -  ,  —  ,  i  ,2,3. 
PARAMETERS. 

Some  selected  plane  A  B  C  is  called  the  parametral  face  or  unit 
face,  and  the  values  a  b  c,  which  express  the  simplest  ratios  of  its 
intercepts,  are  called  the  parameters  of  the  crystal. 

INDICES. 

It  is  always  possible  to  express  the  ratios  between  h,  k  and  /  by 
three  whole  numbers  because,  as  just  stated  they  are  simple 
rational  numbers,  and  if  fractional  they  may  always  be  cleared  of 
fractions  without  changing  the  ratios,  for  evidently 

h  :  k  :  /=  mh  :  mk  :  ml 

The  simplest*  whole  numbers  which  express  the  ratios  of  h,  k 
and  /  are  called  the  indices  .of  the  face,  and  are  always  written  in 
the  same  order,  the  first  referring  to  the  intercept  on  the  axis  X  X, 
the  second  to  that  on  Y  Y,  the  third  to  that  on  Z  Z.  A  bar  over 
an  index,  as  k,  indicates  that  the  intercept  is  negative.  As  in- 
dicating a  face  they  should  be  written  (hkl),  though  frequently 
the  parenthesis  is  omitted  ;  but  if  used  as  a  symbol  of  a  form  they 
should  be  written  \hkl  \. 

Indices  and  intercepts  are  inversely  proportional,  for  from  the 
equation  above  we  obtain  : 


in  which  a,  b  and  c  are  constant  for  all  faces  of  a  crystal. 

If  a  face  is  parallel  to  an  axis  its  intercept  on  that  axis  is  infinite 

and  its  index  is  zero;  for  example,  if  j  =<x>  ,  then  h  =o  and  the 
symbol  is  (o/£/). 

*  Experience  shows  that  the  faces  which  most  frequently  occur  will,  with  proper 
selection  of  the  parametral  plane,  have  as  indices  o  or  I  or  rarely  2. 


GEOMETRICAL  CHARACTERS. 


13 


The  values  of  the  intercepts  OH,  OK,  OL,  become  the  para- 
metral  values  only  when  kkl=ni.  For  with  these  values  only 
can  we  obtain  : 


DETERMINATION  OF  THE  ELEMENTS  OF  A  CRYSTAL, 

The  three  axial  planes  and  the  "  parametral  face  "  constitute  the 
elementary  planes.  These  are  chosen  to  yield  the  simplest  indices 
and  are  usually  planes  of  cleavage,  or  twinning  or  gliding  planes, 
and  are  preferably  planes  of  symmetry. 

In  the  most  general  case  there  are  five  undetermined  elements, 
namely  : 

The  angles  between  the  axes,  Y  Z  or  a,  X  Z  or  /?,  X  Y  or  y. 
The  ratio  between  the  parameters  a^  b,c,  in  which  b  =  I. 

I  °  Determination  of  the  angles  between  the  axes. 
These  may  be  determined  most  simply  from  the  measured  angles 
between  the  axial  planes,  as  follows  : 

Let  the  surface  of  a  sphere,  described  around 
O,  the  centre  of  the  crystal,  meet  the  axes  in 
X,  Y  and  Z,  Fig.  29. 

Construct  A  B  C  the  polar  triangle  of  X  Y  Z, 
then  will  A  be  the  pole  of  the  axial  plane 
YOZ,  B  the  pole  of  XOZ,  and  C  the  pole 
of  XOY. 

The  sides  of  A  B  C  therefore  measure  the 
normal  angles  between  the  axial  planes, 
FIG.  29.  that  is  between  the  planes  (oio),  (100),  (ooi) 

and  these  angles  are  determined  by  measurement 

A  B  =  (loo)  —  (oio),  B  C=  (oio)  —  (ooi),  A  C=(ioo)  —  (ooi). 
The  angles  of  the  triangle  can  therefore  be  calculated  by  formula 


-  cos  BC 
for  instance  :  cos  CAB  = 


cosABcos  AC 


—  :  —  r-  o  —  :  —  ^-~  — 
sin  A  B  sin  A  C 

Since  ABC  and  X  Y  Z  are  polar  triangles 

a=  YZ=i8o°  —  CAB,   /9=XZ=  180°  —  CBA, 
=XY=  180°  —  ACB. 


14  CHARACTERS  OF  CRYSTALS. 

The  angles  between  the  axes  are  therefore  determined. 
EXAMPLE.     In  Axinite  by  measurement  it  is  found  that 

log.  cos.  log.  sin. 

ABor(ioo  —  oio)  =  48°2i'    8"  whence  9.82253     9.873465 
BC  or(oio^ooi)=97°50/    8"       "       9-13459     9-99593 
AC  or  (loo  —  ooi)=  93°  48'  56"       "        8.82311     9.99904 

/cos  C A B=  9.09181 
CAB=82°  54'  1 3"  or  97°    5'  47" 
a  =  97°    5'  47"  or  82°  54'  1 3" 

Similarity  /?  and  Y  can  be  determined 

2°  Determination  of  the  parameter  ratios . 

In  Fig.  29  let  HKL  be  any  plane  cutting  the  three  axes  and 
with  known  indices  likl. 

The  spherical  triangle  rst  described  from  L  as  a  centre  can  be 
solved  because  its  angles  are 

/=(ioo — (oio)  s=  (100)  —  (Jikl]  and  r=  (oio)  —  (hkl),  all  of 
which  may  be  measured. 

The  sides,  tr  and  tst  may  be  calculated  by  formula,  for  instance : 

cos  .y+cos  t  cos  r. 

cos  tr= : — 

sin  t  sin  r 

These  sides  measure  angles  in  the  plane  triangles  H  O  L  and 
K  O  L,  respectively,  and  in  each  of  these  one  other  angle  has  al- 
ready been  determined.  The  triangles  may,  therefore,  be  solved 
for  the  relative  lengths  of  their  sides — that  is,  for  the  intercepts  of 
the  plane. 

For  instance,  in  K  O  L,  the  angle  at  L  is  measured  by  ts,  and 
the  angle  atO  by  Y  Z  or  «,  hence  the  angle  at  K  is  180°— (a  +  ts), 
and  the  sides  are  given  by  the  formula, 

O  L  :  O  K=sin  (180°  —a—ts  )  :  sin  ts 
Similarly, 

O  L  :  O  H  =  sin  (180°— /?— tr  )  :  sin  tr 

EXAMPLE  : 

Given  /=  100°  41',  ^=59°  10',  r=76°  33',  «=82°2i/,  /5=73°  nf. 
Required  a,  b  and  c. 


GEOMETRICAL  CHARACTERS.  15 

cos  J+CQS  /  cos  r_. 5125  +  . I854X. 2335  _ 

sin  /  sin  r  .98215  x  .9726  t9' 

^-54°  35'- 

cos  ?  +  cos  /  cos  s     .2335  +  . I854X. 5125 
COS  ts  =  -       sin  t  sins  -. 98215  X. 8587 

^=67°  5'. 

In  triangle  K  O  L  O  L:  O  K==sin  (180°  —82°  21'  —67°  5'): 
sin  67°  5',=sin  30°  34r :  sin  67°  5^=0.5085  :  .9205  =  ^525  :  I. 

In  triangle  HOL  OL:OH=sin  (180°— 73°  u'_54035'): 
sin  54°35r=sin  52°  14':  sin  54°  35^=0.7905  :  .8150=. 5525  :  .5696 
Hence  O  H  :  O  K:  O  1^.5696:  I  :  .5525=  a-.b'.c. 


CHAPTER  III. 


SPHERICAL  PROJECTION. 


Imagine  a  sphere  described  around  the  centre  of  a  crystal 
with  any  radius  and  radii  drawn  normal  to  each  face. 

The  point  in  which  the  radius  normal  to  any  crystal  face  meets 
the  surface  of  the  sphere,  is  called  the  pole  of  the  face,  and  is  de- 
noted by  the  symbol  of  the  face. 

Planes  which  intersect  in  parallel  edges  will  evidently  have  their 
normals  in  one  plane  and  their  poles  in  the  circle  which  it  cuts 
from  the  sphere  of  projection.  Such  a  series  of  planes  constitute 
a  Zone,  the  plane  of  the  normals  is  the  Zone  Plane,  the  circle  is 
the  Zone  Circle,  and  the  line  through  the  centre  parallel  to  the  face 
and  edges  of  the  zone  is  the  Zone  Axis. 

Fig.  30  represents  the  section  made 
by  a  zone  plane,  the  normals  to  the 
faces  A,  B,  C,  etc.,  meet  the  zone  circle 
in  the  poles  Ap  ~BV  Clt  etc. 

The  same  radii  are  evidently  nor- 
c<  mal  to  the  crystal  faces  of  the  enclosed 
ideal  form  in  which  equivalent  faces 
are  equally  distant  from  the  centre. 
Hence  the  poles  on  the  surface  of  the 
sphere,  in  their  arrangement  will  reveal 
the  hidden  regularity  of  unequally  de- 
veloped crystals. 
Fig.  30  shows,  also,  that  the  arc  of  the  zone  circle  between 
any  two  poles  measures  the  normal  angles,  the  supplements  of  the 
angles  between  the  corresponding  faces.  Different  intersecting 
zone  circles  give  spherical  triangles,  the  sides  of  which  are  normal 
angles,  which  can  be  solved  by  simple  formulae,  provided  certain 
parts  have  been  determined  by  measurement  or  previous  calcula- 
tion. 


GEOMETRICAL  CHARACTERS.  17 

A  number  of  simple  relations  between  the  faces  in  zones  have 
been  deduced  by  means  of  which  it  is  possible  to  greatly  reduce 
the  number  of  necessary  direct  measurements  and  to  simplify  the 
calculations. 

SYMBOL  FOR  A  ZONE  Axis. 

The  direction  of  intersection*  of  two  crystal  faces  (Jikt)  and 
(li1  k'  I1)  is  [uvw]  in  which  u  =  kl'  —  Ik',  v  =  Ik'  —  hi1 ,  vr^htf—Mi'. 

These  values  may  be  obtained  by  cross  multiplication  of  the 
twice  written  indices,  striking  off  end  terms  and  reading  down 
alternately  from  left  to  right  and  from  right  to  left,  thus  : 


k     I      k      k 

XXX 
k'    I'     h'  -   k' 


As  all  the  terms  are  whole  numbers  the  values  of  u,  v  and  w  will 
be  also.  A  zone  is  designated  by  this  symbol  [uvw]  or  by  the 
symbols  of  two  of  its  planes  \Jikl,  pqr^y  or  by  letters  designating 
the  faces  [P,  Q],  always  enclosing  with  the  parallel  bars. 

EQUATION  FOR  ZONE  CONTROL  OR   CONDITION  THAT  A  FACE  MAY 
BELONG  TO  A  ZONE. 

If  a  face  (pqr)  lies  in  a  zone  [uvw]  its  indices  must  satisfyf  the 
equation  /u  +  qv  -\-  wr  =  o. 

If  two  indices  of  a  face  are  known,  and  the  zone  is  known,  the 
third  index  may  therefore  sometimes  be  found. 

Example.  By  test  with  a  reflection  goniometer  the  face  (3/£i)  is 
found  to  be  in  the  zone  [mj;  substituting  in  the  above  equation 
^iX  i  =  o  whence  3  —  k  —  i  =  o  and  k=  2. 


FACE  IN  TWO  ZONES. 

The  indices  of  a  facej  which  lies  in  the  zones  [uvw]  and  [u'v'w'] 
will  be  h  =  uv'  —  vu',  k  =  vw'  —  wv',  /=  wu'  —  uw'. 

These  values  may  be  obtained  by  cross  multiplication. 

Example.  —  By  test  a  face  (kkl)  is  found  to  lie  in  the  zones 
[  2!  o  ]  and  [  o  i  2  ],  required  the  values  of  //,  k  and  /. 

*  Miller's  Treatise  on  Crystallography  ,  p.  7. 
•j-  Ibid  p.  10. 
J  Ibid,  p.  8. 


i8 


CHARACTERS  OF  CRYSTALS. 
o 


I       O      2       I 

XXX 


=  2  —  O,  £=0-1-4,  /  =  2  —  O, 
I       2      O      I 

that  is,  (hkl)  =  (242)=  (121). 

To  FIND  A  FOURTH  FACE  IN  A  ZONE.* 

Let  A  =  (efg),  B  =  (/£/£/)  and  C  =  (pqr}  be  known  faces 
of  a  zone,  the  poles  of  which  lie  in  the  order  named. 

To  find  the  position  of  any  fourth  face,  D  =(mno},  if  its  in- 
dices are  known. 


AC—  cot  A  D== 


AB  —  cot  AD). 


AC          A  B 

The  values  of  ^-^  and  ^-=?  are  such  as  result  by  cross  multi- 

(^f    .L/  -D    J_/ 

plication  of  pairs  of  corresponding  indices  : 


V     / 
X 


/   g 
X 


p     q       eq—fp     .         q     r      fr—gq  . 

—»  V*  —  =  —  —  > 


CD       p     q      pn  —  qm 

X 
m   n 

'    S 
X 

or    <- 


X 


qo  —  rn 


p     r        p  o  —  rin 

X 
m     o 


e     f 

X 
AB      h     k         ek—fh 


f   g 

X 
k     I  _fl—gk 

IT^l^  ko  —  ln; 

X 
n     o 

e    g 

X 
h     I  _     el—gh 

h     I       ho  —  Im 

X 
m    o 

Very  frequently  two  of  the  three  identical  ratios  are   indeter- 
minate. 


BD      h     k~  hn  —  fan 
X 


or 


*  Groth's  Fhysikalische  Krystallographie^  III.  ed.,  p.  584. 


GEOMETRICAL  CHARACTERS.  19 

EXAMPLE. 

In  a  crystal  of  pyroxene,  given:  A=  (efg]  =  (100) ,  B  =  (likl) 
=  (101),  C=(/^=(ooi),  D=(w;w)=(3~oi).,  AB=49°  39', 
A  C  =  73°  59'.  Required  A  D. 

By  trial  the  first  and  third  ratios  are  found  to  be  indeterminate, 
but  from  the  third  we  obtain 

AC       i.i — o.o      i     AB       i.i — o.i       I 


CD      o.i  — 1.3      3     B  D       i.i  — 1.3      4 

substituting  in  the  equation 

i  (cot  73°  59'  —  cot  AD)=-(cot49°  39' —  cot  AD) 

1.1504  —  4  cot  AD—  2.5487—3  cot  AD.   cot  AD=  1.3983 

AD=  144°  26' 
ZONE  OF  TWO  PINACOIDS. 

Every  face  in  the  zone  will  have  that  index  zero  which  is  zero 
in  both  pinacoids.  For  example  if  a  face  lies  in  the  zone  of 
[100,  oio]  its  third  index  must  be  zero  for  the  zone  symbol  is 
[ooi]  p.  17  and  h.  o  -f  k.  o  -f  /.  I  =  o  can  be  true  only  if  /—  o. 

ZONE  THROUGH  ONE  PlNACOID. 

The  ratio  of  the  two  indices  which  are  zero  for  the  pinacoid,  is 
constant  for  all  faces  of  the  zone.  For  example  the  symbol  of  the 
zone  [\oo,hkl~\  is  \plk~\.  Any  third  face  (pqr)  must  satisfy  the 

k       a 
equation  p&  -\-  q  I—  r  £=  o,  that  is  q  1=  rk  or  -=-  =  ~ 

ZONES  IN  WHICH  TWO  INDICES  ARE  CONSTANT. 

If  two  faces  have  two  corresponding  indices  in  each  with  the 
same  ratio,  all  faces  in  their  zone  will  have  those  two  indices  in 
that  ratio.  For  example,  the  symbol  of  the  zone  of  the  faces  (123) 
and  (245)  is  [210]  A  face  (hkl)  to  be  in  this  zone  must  satisfy 
the  equation  —  2/*-f/£-fo/=o,  that  is  2  h  =  k. 

CHANGING  AXES.* 

If  three  edges  are  preferred  to  those  originally  selected  as  axes 
directions  proceed  as  follows  : 

From  the  original  indices  of  the  new  axial  planes  determine  the 
symbols  of  their  intersections  (that  is,  their  zone  symbols),  [uvw], 
[ulVlwJ  [u2v2w2].  ______ 

*  Miller  Treatise  on   Crystallography,  p.  17. 


20  CHARACTERS  OF  CRYSTALS. 

Then,  if  the  indices  of  any  face  referred  to  the  old  axes  is  (hki), 
its  new  indices  h^k^  and  ^  will  have  the  following  values: 

hv  —  /m  -f  hv  -f  //w 
kl  —  kuv  -f-  ^Vj  +  /7w1 
/t  =  /u2  +  /v2  +  /w2. 

CHANGING  PARAMETERS. 

hkl  the  indices  of  a  face  referred  to  parameters  <z,  £,  <:, 


"         "       ""  "         " 


JL=.fL,   ^J_  =  —  ,  _J  !  =  _  then  if  this  face  is  chosen  as  the  parame- 
//j        h     k^  /!        / 

tral  plane  al=~,  bl=  —,  ^  =  -^  and  any  other  face,  the  indices 

It  K  I 

of  which  were/^r,  will  receive  new  indices  p^q^  in  which 


STEREOGRAPHIC  PROJECTION. 

The  sphere  containing  the  poles  of  the  crystal  faces  is  most 
conveniently  represented  in  Stereographic  Projection. 

Some  important  plane  through  the  centre  of  the  sphere  is  se- 
lected as  a  plane  of  projection,  and  all  poles  of  the  sphere  are 
projected  in  this  plane  and  fall  within  the  limits  of  the  so-called 
"  PRIMITIVE  CIRCLE." 

For  instance,  let  Fig.  31  represent  the  upper  half  of  a  crystal  of 
cassiterite  within  a  sphere  of  projection.  The  crystal  faces  a,  d, 
g,  k,  m.,  etc.,  will  have  their  poles  in  the  points  A,  D,  G,  klf  mlt 
etc.,  where  the  normal  radii  intersect  the  spherical  surface. 

Let  the  equatorial  plane  be  the  plane  of  projection,  let  the  south 
pole  be  the  point  of  sight,  then  if  lines  are  drawn  from  the  south 
pole  to  the  various  poles  in  the  northern  hemisphere,  they  will  all 
pierce  the  equatorial  plane  at  points  A,  D,  G,  K,  M,  etc.,  in  the 
primitive  circle  which  are  the  projections  of  the  poles. 

PRINCIPAL  CHARACTERISTICS  OF  STEREOGRAPHIC  PROJECTION. 
1°  All  zone  circles  at  right  angles  to  the  primitive  circle  are 


GEOMETRICAL  CHARACTERS. 


21 


FIG.  31. 

projected  as  diameters,  all  others,  as  arcs  of  circles,*  cutting  the 
primitive  circle  in  the  extremities  of  a  diameter. 

2°  If  the  pole  (F.  Fig.  jj)  of  a  zone  circle  CPCV  is  united  with  the 
poles  of  two  faces  (P  and  Q)  of  the  zone  by  straight  lines,  and  these 
prolonged  to  the  circumference,  they  cut  from  it  an  arc  equal  to  the 
normal  angle  between  the  two  faces. "\ 

Free-hand  construction  in  many  instances  serves  to  give  a 
comprehensive  view  of  the  symmetry  and  relations  of  the  faces 
and  zones.  For  accurate  construction  a  few  rules  are  needed, 
which  are  based  upon  characteristics  I  and  2. 

PROBLEM  i. 

Given  the  projection  of  a  zone  circle,  to  find  tJiat  of  its  pole. \ 

*For  demonstration,  Story-Maskelyne's  Crystallography,  p.  30. 

f  Ibid,  p.  33. 

\  Miller's  Treatise  on  Crystallography,  p.  133. 


22 


CHARACTERS  OF  CRYSTALS. 


FIG.  32. 


In  Fig.  32.     Given  C  P  Q  to  find  F. 

C  P  Q  intersects  the  primitive  circle  in 
the  diameter  C  Cr  Its  pole  must  lie  on 
that  circle  which  is  projected  as  the 
diameter  D  D4  normal  to  C  Q  and  at  90° 
from  the  point  T  where  the  two  circles 
'intersect.  Draw  C  E  through  T,  layoff 
EE^  90°,  draw  ££,  then  is  F  the  re- 
quired pole  because  it  is  a  point  on  D  Dl 
and  by  the  second  characteristic  it  lies  at 
a  quadrants  distance  from  T.  . 


PROBLEM  2. 

Given  the  projection  of  the  pole  of  a  great  circle  to  draw  the  circle. 

In  Fig.  32  given  F  to  find  C  P  Q. 

Draw  a  diameter  D  Dt  through  F  and  another  C  Q  normal  to 
this.  Through  F  draw  CEj,  make  EEj  90°,  draw  EC,  then  are 
T,  C  and  Clf  three  points  of  the  zone  desired.  Through  these  pass 
the  arc  of  a  circle. 

PROBLEM  3. 

In  any  zone  given  the  angle  between  two  faces  and  the  projection  of 
one  of  them  to  find  that  of  the  other. 

In  Fig.  33,  given  the  zone  CPQ,  the  face  P,  and  the  angle  P  to 
Q=a.  To  find  Q, 


FIG.  33. 


FIG.  34. 


Find  F  the  pole  of  the  zone  C  PCj,  as  in  problem  I.  Draw  FE 
through  P,  make  EEj  equal  a,  and  draw  EjF.  Then  by  the  second 
characteristic,  Q  is  the  desired  projection. 

In  Fig  34,  is  represented  the  special  case  in  which  the  given 


GEOMETRICAL  CHARACTERS. 


zone  C  P  Qis  normal  to  the  primitive  circle.     In  this  case  the  pole 
of  the  zone  falls  at  D,  the  extremity  of  the  diameter  normal  to  C  Q. 
If  a=i8o°,  the  second  face  will  lie  outside  the  primitive  circle. 
The  construction  is  not  changed. 

PROBLEM  4. 

Given  the  projections  of  tivo  faces 
tn  a  zone  to  find  the  projection  of  the 
zone  circle* 

In  Fig.  35,  given  P  and  Q  to 
find  C  P  Q. 

Draw  a  diameter  through  P.  The 
face  P!  opposite  P,  will  be  on  it. 

Draw  O  A  perpendicular  to  this 
diameter,  draw  P  A,  and  from  A,  a  FlG'  35< 

line  perpendicular  to  P  A.  The  intersection  of  this  perpendicular 
with  the  diameter  through  P,  will  be  Px ;  the  projection  of  the  pole 
of  a  face  opposite  P,  that  is  1 80°  from  P,  in  the  same  zone. 

The  arc  of  a  circle  through  the  three  points,  P,  Q  and  Plf  will  be 
the  desired  circle. 

EXAMPLES.     Application  of  preceding  constructions  to  the  pro- 
jection (Fig.  36)  of  the  cassiterite  crystal  (Fig.  31). 


The  poles  lying  in  the  primitive  circle  are  separated  by  arcs 

*Miller's  Treatise  on  Crystallography,  p.  133. 


24 


CHARACTERS  OF  CRYSTALS. 


equal  to  the  true  normal  angles.  The  poles  of  all  other  faces  in 
this  instance  lie  in  zone  circles  normal  to  the  primitive  circle, 
therefore  projected  as  diameters.  The  true  position  of  any  one  of 
these  is  found  by  Problem  3.  For  example:  M — J  or  (no)  — 
(Hi)  =  46°  27'.  Lay  off  J  x  =  46°  27',  draw  x  I  to  the  pole  I 


FIG.  37. 


FIG.  38. 


of  the  circle  J  H  cutting  the  latter  at  M,  the  required  projection. 
The  pole  R  is  in  the  zone  of  M  and  J,  and  being  an  equivalent  face 
O  R  =  O  M. 

Draw  zone  circles  through  C  M  D,  B  M  A,  C  R  D  and  B  R  A. 

The  poles  of  N  and  S  will  lie  at  the  intersections  as  shown. 
For  the  second  order  forms  K  lies  in  the  zone  [A  B]  and  in  the  zone 
[M  N],  hence  is  at  K.  Similarly  the  faces  e,  I  and/ have  poles  E, 
L  and  F. 

A  crystal  of  barium  chlorate  shown  in  Fig.  37  yields  the  pro- 
jection Fig.  38.  The  plane  of  projection  is  taken  normal  to  the 
zone  [m,  a]. 

Given;//  —  a  =  48°  53j£',/«  —  ^=41°  51^', '  m—p=  56°  39' 
In  Fig.  37  lay  off  arcs  equal  48°  53^'  each  side  of  a  and  a'  de- 
termining m,  m',  m",  m"  ' .  Lay  off  an  angle  equal  41°  51^'  and 
draw  a  line  (not  shown)  to  pole  of  the  zone  \a  a'~\,  intersecting  the 
latter  at  r.  By  trial  /  lies  in  the  zone  \_m" '  r~\f  construct  this  zone 
circle,  find  its  pole,  F  by  Problem.  Lay  off  mf  s  =  56°  39',  draw 
s  F,  then,  by  Problem  3,/  is  the  desired  pole;  p'  is  in  the  zone  \in  r\ 
symmetrical  to  /. 


CHAPTER  IV. 


THE  THIRTY-TWO  CLASSES  OF  CRYSTALS. 


In  this  classification,  following  Professor  Groth,  the  conception 
of  hemihedral  and  tetartohedral  forms  is  abandoned  because  the 
geometrically  connected  whole  and  partial  forms  have  no  structural 
connection  and  are  incapable  of  occurrence  upon  crystals  of  the  same 
substance. 

Each  occurring  form  is,  therefore,  considered  to  be  complete  and 
independent,  and  with  a  grade  of  symmetry  which  is  not  in  every 
case  determinable  from  a  consideration  of  the  grouping  of  the  faces, 
but  involves  the  far  wider  conception  that  two  directions  are  not 
structurally  equivalent  unless  they  are  equivalent  with  respect  to  all 
properties,  and  that  two  forms  geometrically  identical  are  not  struc- 
turally so  if  in  corresponding  directions  there  is  revealed  an  essen- 
tial difference  in  behavior  with  polarized  light  or  etching  or  pyro- 
electricity  or  any  other  test,  the  results  of  which  depend  upon  the 
manner  the  crystal  molecules  are  built  together. 

With  this  conception  the  same  geometric  form  may  occur  on 
crystals  structurally  different,  and  is  to  be  regarded  as  in  each  case 
a  limit  form  of  geometrically  distinct  general  forms.  If  a  class  be 
made  of  each  conceivable  variation  of  general  form  and  its  geo- 
metric limit  forms,  two  great  essentials  will  be  fulfilled : 

1.  All  known  and  some  unknown  forms  will  be  classed. 

2.  Each  class  will  consist  of  forms  capable  of  occurring  upon- 
crystals  of  the  same  substance,  and  will  contain  all  the  forms  which 
can  occur  on  these  crystals. 

There  have  been  distinguished  thirty-two  grades  or  classes  of 
symmetry  which  may  be  united  into  six  systems  by  grouping  to- 
gether classes  in  which  the  selected  axes  of  reference  are  geo- 
metrically similar. 


26 


CHARACTERS  OF  CRYSTALS. 


In  the  stereographic  projection  of  the  general  form  given  under 
each  class  all  full  lines,  whether  diameters,  circles  or  arcs  of  cir- 
cles represent  planes  of  symmetry  and  the  small  black  ellipses, 
triangles,  squares  and  hexagons  represent  the  axes  of  binary, 
ternary  quaternary  and  senary  symmetry. 


TRICLINIC  SYSTEM. 

This  system  must  include  all  crystallographic  forms  which  can 
only  be  referred  to  three  non-equivalent  axes,  Fig.  39,  at  oblique 
angles,  a,  ft,  and  fy  to  each  other. 

The  selection  of  axes  is  arbitrary,  two  edges  are  chosen  as  direc- 
tions of  the  basal  axes,  a  and  b.  Two  planes  from  the  zones  of 
these  edges  are  chosen  as  Jiooj  and  Joioj  and  their  intersections 
are  the  vertical  axes  c. 


c 


X/      X 


FIG.  39. 


X. 

/  \j 

\ 


a 
FIG.  40. 


i.  UNSYMMETRICAL  CLASS. 

Without  either  planes  or  axes  of  symmetry.  The  projection, 
Fig.  40,  shows  that  the  symmetry  of  the  class  is  satisfied  with  one 
face  for  the  most  general  form.  EXAMPLE. — Calcium  thiosulphate, 
CaS2O3-6H2O. 

2.  CLASS  OF  PINACOIDS. 

With  composite  symmetry  to  a  binary  axis  and  a  plane  normal 
thereto. 

As  shown  in  projection,  Fig.  41,  any  upper  face  indicated  by  X 
would  by  simultaneous  rotation  around  the  axis  and  reflection  in 
the  plane  coincide  with  the  diametrically  opposite  lower  face,  that 
is,  the  symmetry  of  the  class  is  satisfied  by  two  faces  for  the  most 


GEOMETRICAL   CHARACTERS. 


27 


general  form  or  PINACOID  (Tetarto  Pyramid),  Fig.  42.   EXAMPLES. — 
Albite,  cyanite  and  chalcanthite. 


o 


a 
FIG.  41. 


FIG.  42. 


THE  Six  LIMIT  FORMS. 

In  each  class  there  are  six  limit  forms  corresponding  to  six 
special  positions  of  the  faces  of  the  general  form.  In  Class  2 
these  will  be  pairs  of  parallel  planes,  and  in  Class  I  single  planes 
only. 


010 


Position  of  any  Face.     Symbol. 

1.  Parallel  to  a  and  b.         JooiJ 

2.  Parallel  to  a  and  c. 

3.  Parallel  to  b  and  c. 

4.  Parallel  to  a. 

5.  Parallel  to  b. 

6.  Parallel  to  c. 


Class  i. 

PLANE  of  Fig.  43. 
PLANE  of  Fig.  44. 
PLANE  of  Fig.  45. 
PLANE  of  Fig.  46. 
PLANE  of  Fig.  47. 
PLANE  of  Fig.  48. 


Class  2. 

PINACOID  Fig.  43. 
PINACOID  Fig.  44. 
PINACOID  Fig.  45. 
PINACOID  Fig.  46. 
PINACOID  Fig.  47. 
PINACOID  Fig.  48. 


FIG.  43. 


FIG.  44. 


FIG.  45. 


28 


CHARACTERS  OF  CRYSTALS. 


FIG.  46. 


FIG.  47. 


FIG.  48. 


PROJECTION  AND  CALCULATION  OF  TKICLINIC  FORMS. 

THE  PROJECTION  is  conveniently  made  upon  a  plane  normal  to 
the  vertical  axis  c.  All  planes  in  the  zone  of  c  as  (100),  (oio); 
(no),  being  projected  in  the  primitive  circle  at  the  measured 
angles  apart. 

The  pole  of  any  face  P  will  lie  at  the  intersection  of  two 
circles  found  as  follows  :  *  Their  centers  are  on  prolongations 
of  diameters  through  ^(100)  and  J5(oio)  and  distant  from  the 

¥  Y  ' 

center  0  of  the  primitive  circle  respectively  OK=-—,  OL  =  -^ 

and   their   radii   are   respectively  Kp^rtanPA,  Lp  =  rtenPB. 

Most  of  the  poles  are  found  by  zones  and  problem  3,  p.  22. 

THE  CALCULATIONS  are  principally  solutions  of  spherical  tri- 
angles t  and  of  equations  which  connect  measured  angles  with 

*Groth,  Physikalische  Krystallographie,  p.  580,  III.  ed. 
\  For  right-angled  spherical  triangles 

C=  90°         a,  b,  c  =  sides  opposite  angles  A,  B,  C. 

sin  a        cos  B  tan  b 

cos  A  =  —  -7  =cos  a  sin  B, 


sin  A 


cos  b  ' 


tanr 


tan  a 
tan  A  =  -  —  T-»  cos  c  =  cos  a  cos  b  =  cot  A  cot  B 

In  oblique  angled  spherical  triangles 
A,  B,  C  denote  angles  ,    a,  b,  c,  opposite  sides,     _IL  _  J  _  =  $t    an& 


sin  A 
sin  a 


sin  B       sin  C 


sin  b        sin  c 
cos  A  =    —  cos  B  cos  C  -f-  sin  B  sin  C  cos  # 
cos  a  =         cos  <5  cos  c  -f-  sin  £  sin  r  cos  ^4 


2 

A 


sin  b  sin  c 

sin  s  sm  (s — a] 
sin  b  sin  r 


4 


*L.  =  -t  /  — cos  ^ cos  ( s — ^ ) 

2 


sin  ^  sin  6' 


GEOMETRICAL  CHARACTERS. 


29 


indices  and  elements.     Wherever  possible  the  zonal  equations,  pp. 
16-20,  are  used  to  simplify  the  calculations. 

DETERMINATION  OF  ELEMENTS. 

The  general  determination  of  pp.  13-15  requiring  the  meas- 
urement of  five  angles  between  four  planes,  no  three  of  which  lie 
in  the  same  zone,  is  followed  as  in  the  example  there  given. 

GENERAL  EQUATION  BETWEEN  AXES  AND  INDICES. 

Let  any  plane  the  indices  of  which  are  hkl  meet  the  axes  in 
HKL,  let  the  axes  meet  the  surface  of  a  sphere  described  around 
O,  in  XYZand  let  the  normal  Op  to  HKL  meet  the  sphere  in  P. 
Fig.  49. 

From  a  section  through  OHp  it  is  evident  that  Op=OH 
cos PX, and, similarly,  Op=OK cos PFand  Op=OL cos PZ.  Equa- 
ting, OH  cos  PX=OKcos  PY=OL  cos  PZ. 

•From  page  13,  we  have 


ffl-, 


Substituting, 


PZ 


FIG.  49. 


To  DETERMINE  POSITION  OF  ANY  POLE  P. 

Let  Fig.  50  show  the  poles  in  projection.     From   the  following 
equations  the  position  of  P  will  result  if  the  indices  and  elements 


.  30  CHARACTERS  OF  CRYSTALS. 

•are  known,  or,  conversely,  the  indices  will  result*  if  the  position  is 
known  in  one  zonal  circle. 

sin  AB sin  CH __  sin  CAP  _ck 
sin  CA  smBH~  sin£AP  ~~bl 

sin  BCsm  AK_sm  ABP_  al 
sin  AB  sin  CK~'Sri~CBP~~ck 

sin  CA  sin  BL__sm  BCP_ bh 
sin  JSCsin  AL~~s'm  ACP~ak 

If  any  five  of  the  arcs  in  the  following  are  known,  the  sixth 
results 

sin  BHsm  £ATsin  AL=sm  CHsm  AKsm  BL. 

IF  THE  ELEMENTS  ARE  UNKNOWN  the  indices  of  a  face  lying  in  a 
known  zone  may  be  determined  by  measuring  the  angle  to  one 
face  in  that  zone,  and  as  three  faces  must  already  be  known,  sub- 
stituting in  the  formula  to  find  fourth  face  in  a  zone,  p.  18. 
Conversely,  if  the  indices  of  such  ajplane  are  known,  its  position 
may  be  calculated. 

To  FIND  t  THE  ARC  JOINING  ANY  TWO  POLES  P  AND  P' . 

Calculate  by  preceding  equations  the  distances  of  P  and 
P1  from  one  of  the  poles  A,  B  or^C'and  the  angles  that  these 
distances  make  with  one  of  the  adjacent  sides  of  ABC,  therefore, 
the  angle  that  they  make  with  each'other.  From  two  sides  and 
included  angle  calculate  the  third  side  PP. 

MONOCLINIC  SYSTEM. 

All  forms  in  this  system  must  be  referable  to  three  non-equiva- 
lent axes,  Fig.  51,  two  oblique  to  each  other,  the  third,  normal  to 
their  plane. 

Conventionally  the  normal  or 
ortho  axis  b  extends  from  right 
to  left,  either  of  the  other  axes 
is  made  the  vertical,  c,  and  the 
third  the  clino,  a,  dips  down- 
ward from  back  to  front.  The 
acute  angle  between  the  ver- 
tical and  clino  axes  is  /9.  FIG.  51. 

*  Story-Maskelyne's  Crystallography,  p.  430. 
f  Miller's   Crystallography^.  98. 


GEOMETRICAL  CHARACTERS. 


The  system  comprises  three  classes,  in  which  are  possible  many 
series  each  including  all  the  forms  which  can  be  referred  to  the 
same  value  for  /?  and  to  the  same  parameters  a,  b%  c. 

3.  CLASS  OF  THE  MONOCLINIC  SPHENOID. 

With  one  axis  of  binary  symmetry.  As  shown  in  the  projection 
upon  (oio)  Fig.  52,  the  pole  of  any  face  by  rotation  1 80°  around 
the  binary  axis  must  reach  the  pole  of  an  equivalent  face.  The 
two  faces  satisfy  the  symmetry  for  the  most  general  form  or  SPHE- 
NOID, Fig.  53.  EXAMPLES. — Tartaric  acid,  milk  sugar. 


FIG.  52. 


FIG.  53. 


4.  CLASS  OF  THE  MONOCLINIC  DOME. 

With  one  plane  of  symmetry.  As  shown  in  the  projection  of  the 
general  form,  Fig.  54  any  face  reflected  in  the  plane  of  symmetry 
coincides  with  an  equivalent  opposite  face.  The  two  faces  satisfy 
the  symmetry  of  the  class  for  the  most  general  form  or  DOME,  Fig. 
55.  EXAMPLE. — Potassic  tetrathionate  K2S4O6. 


FIG.  54. 


FIG.  55. 


5.  PRISMATIC  CLASS. 

With  one  plane  of  symmetry,  Fig.  56,  at  right  angles  to  an  axis 
of  binary  symmetry.     As  shown  in  projection,  Fig.  57,  any  face 


32  CHARACTERS  OF  CRYSTALS. 

the  pole  of  which  is  marked  X  by  rotation  of  180°  around  the  bi- 
nary axis  coincides  with  an  equivalent  face  in  the  alternate  octant, 
these  reflected  in  the  plane  of  symmetry  coincide  with  faces  the 
poles  of  which  are  marked  by  a  circle. 


FIG.  56. 


FIG.  57. 


FIG.  58. 

That  is  four  planes  satisfy  the  symmetry  for  the  most  general 
form  or  PRISM,  Fig.  58.  EXAMPLES. — Pyroxene,  orthoclase,  gyp. 
sum. 

THE  Six  LIMIT  FORMS. 

In  each  class  there  are  six  limit  forms  corresponding  to  special 
positions  of  the  faces  of  the  general  form.  These  may  be  tabu- 
lated as  follows : 


Position  of  Any  Face  and  its  Pole. 


1.  Parallel  to  a  and  b 

Poles    projected*    on     the 

primitive  circle 

2.  Parallel  to  a  and  c  .    .    .    . 
Poles  projected  at  centre  .  . 

3.  Parallel  to  b  and   c  .    .    .    . 
Poles     at     intersections      of 

primitive   circle   and   hori- 
zontal diameter  ..... 

4.  Parallel  to  a   Poles   are   on 

diameter  from  (ooi)  .  .    . 


5.  Parallel '  to   b  Poles   are   on 

primitive  circle. 

6.  Parallel   to   c   Poles  are   on 

Horizontal  diameter  . 


Symbol. 


001 


Joio} 


ioo 


\«M\ 


Name  of   Form. 


Classes  to  which 
Form  Belongs. 


PlNACOID,   Fig.  59. 
(Basal  Pinacoid~) 
BASAL  PLANE. 
One  face  of  Fig.  59. 
PINACOID,  Fig.  60. 
(Clino  Pinacoid. ) 
PLANE,  One  face  of  Fig.  60. 
PINACOID,  Fig.  6r. 
(Ortho  Pinacoid.) 
PLANE,  One  face  of  Fig.  61. 

PRISM,  Fig.  62. 
(Clino  Dome.) 
DOME,  Fig.  65. 
SPHENOID,  Fig.  67. 
PINACOID,  Fig.  63. 
(Henri  Ortho  Dome.) 
PLANE,  One  face  of  Fig.  63. 
PRISM,  Fig.  64. 
(Monoclinic  Prism.) 
DOME,  Fig.  66. 
.SPHENOID,  Fig.  68. 


3»  —i     5- 

—  >     4,  — • 

— ,    4,     5- 

3»  — .  — • 
3.  —  >    5- 

—i    4,  — . 

— >  —     5- 

—  4,  — • 
3.  — ,  — • 
3.  — ,    5- 


-,     4,  — • 


*  Plane  of  projection  the  pinacoid  (010). 


GEOMETRICAL   CHARACTERS. 


33 


FIG.  59. 


— >— -4 


FIG.  60. 


FIG.  61. 


FIG.  62.  FIG.  63.  FIG.  64. 

LIMIT  FORMS  OF  CLASS  5. 


FIG.  65.  FIG.  66.  FIG.  67.  FIG.  68. 

OTHER  LIMIT  FORMS  WHICH  ARE  NEW  SHAPES. 

PROJECTION  AND  CALCULATION  OF  MONOCLINIC  FORMS. 
The  plane  of  projection  may  be  the  pinacoid  (oio),  which  is 
a  plane  of  symmetry  in  classes  4  and  5,  when  the  poles  of  all 


34 


CHARACTERS  OF  CRYSTALS. 


planes  in  the  zone  of  b  will  lie  in  the  primitive  circle  as  given  in 
preceding  table.  Or  the  projection  may  be  made,  as  in  the  tri- 
clinic  system,  upon  a  plane  normal  to  the  vertical  axis,  in  which 
case  the  poles  of  planes  in  the  zone  of  b  will  be  projected  on  the 
vertical  diameter  as  described  in  example,  p.  24. 

The  poles  will  be  found  as  in  the  triclinic  system. 

When  a  plane  of  symmetry  appears  as  a  diameter  in  the  pro- 
jection, all  poles  and  indices  will  be  symmetrical  to  this  diameter. 

DETERMINATION  OF  ELEMENTS. 

In  Fig.  29,  p.  13,  B  and  Y  will  coincide  and  BC,  AB  and  / 
become  90°.  Then  AC— ft.  In  the  spherical  triangle  rst, 

OL        cos  ts 


cos  tr  =  - 


cos  s 


costs   = 


cos? 


sin  r 


sin  s  '  OK       sin  ts 


=  cot  ts 


OL       s\n(iSo°-AC—  tr)      Tr   _ 

— gj — -. —     —     If  P   is   the    parametral    plane, 


OH:  OK:  OL  =  a\b\c,  if  not,  a  =  OHJi,  b  =  OK.k,  c  =  OL.!.   That 
is  three  angles  suffice  for  the  determination  of  the  elements. 

To  DETERMINE  THE  POSITION  OF  ANY  POLE  P. 
Using  notation  of  Fig.  69, 

r  7  ~-^ 


b 
.jcot/3 

sin  CK 


a 


CK 


j  sin  AK; 


sin  AK 
k      sin  AD       tan  OB 


ch  %  h  =  sin  AD     sinCK 
~al '  T  ~  sin  CD  '    s'mAK* 


-j  (  cot  AD—  cot  AC)  ;  cos  PA  =  sin  PB 
cos  AK\  cos  PC=  sin  PB   cos  CK 


tan  PB  =  ~r 


l  s'mAD 


k  sinAK 


tan  OB. 


Aioo 


FIG.  69. 


Because  B  is  pole  of  zone  circle  AA't  CK=  PBC,  AK  =  PBA 
and  CK=  i8o°—(P£A  +  CA'),  that  is  the  sines  and  cosines  of 
these  angles  may  be  substituted  for  those  of  the  arcs  in  any^of  the 
formulae  above. 


GEOMETRICAL   CHARACTERS. 


35 


To  FIND  THE  ARC  JOINING  TWO  POLES,  PandP'. 
Proceed  as  in  triclinic,  p.  30. 


ORTHORHOMB1C   SYSTEM. 
All  forms  in  this  system  must 
be  referable  to  three   nonequiva-  c 

lent  axes,  Fig.  70,  at  right  angles 
to  each  other.  The  three  axes 
are  physically  of  equal  impor- 
tance, any  one  may  be  chosen  as 

c,  the  vertical ;  the  longer  of  the   

other  two  will  be  the  macro  or  b  ^^ 

axis ;  the  shorter  axis  (from  front 
to  back)  the  brachy  or  a  axis. 
There  are  as  many  series  of  forms 
possible  as  there  are  irrational 


values  for  -7  and  T» 
b  b 


FIG.  70. 


6.  CLASS  OF  THE  RHOMBIC  BISPHENOID. 

With  three  axes  of  binary  symmetry  at  right  angles  to  each 
other.  As  shown  in  the.  projection,  Fig.  71,  four  faces  satisfy  the 
symmetry  of  the  most  general  form  or  RHOMBIC  BISPHENOID,  Fig. 
72:  EXAMPLE. — Epsomite. 


\. °^ X 

a 


FIG.  71. 


FIG.  72. 


7.  CLASS  OF  THE  RHOMBIC  PYRAMID. 

With  two  planes  of  symmetry  perpendicular  to  each   other  and 
intersecting  in  an  axis  of  binary  symmetry.     As  shown  in  the  pro- 
jection, Fig.  73,  four  faces  satisfy  the  symmetry  of  the   most  gen 
eral  form  or  RHOMBIC  PYRAMID,  Fig.  74.     EXAMPLES. — Calamine, 
struvite. 


*  Story-Maskelyne's  Crystallography,  p.  436  and  Groth's  Phys.  Kryst.,  p.  578. 


CHARACTERS  OF  CRYSTALS. 


-ib 


a 

FIG.  73. 


FIG.  74. 


8.  CLASS  OF  THE  RHOMBIC  BIPYRAMID. 

With  three  planes  of  symmetry  at  right  angles  to  each  other 
which  intersect  in  three  axes  of  binary  symmetry.  These  are 
shown  in  Fig.  75,  and  the  planes  divide  space  into  eight  octants, 
shown  in  projection,  Fig.  76,  as  four  trirectangular  spherical  tri- 
angles. 

Any  upper  face  corresponding  to  a  pole  x  in  the  projection 
would,  by  rotation  of  180°  around  the  vertical  binary  axis,  coincide 
with  an  upper  face  in  the  alternate  octant,  these  reflected  in  the 
vertical  symmetry  planes  coincide  with  two  other  upper  faces  and 
the  four  reflected  in  the  horizontal  plane  coincide  with  four  lower 
planes  the  poles  of  which  are  marked  by  circles.  Since  ft,  k  and  / 
retain  a  constant  order,  there  can  be  sign  permutations  only  cor- 
responding to  one  plane  in  each  octant. 

The  symmetry  of  the  class  is  therefore  satisfied  by  eight  faces 
for  the  most  general  form  or  RHOMBIC  BIPYRAMID,  Fig.  77.*  EX- 
AMPLES.— Aragonite,  marcasite,  barite. 


U-" 


FIG.  75. 


FIG.  76. 


FIG.  77. 


*There  may  be  many  different  pyramids  in  a  series  with  rational  indices  hkl  which 
may  be  all  equal,  any  two  equal  or  all  unequal. 


GEOMETRICAL   CHARACTERS. 


37 


THE  Six  LIMIT  FORMS. 

In  each  class  there  are  six  limit  forms  corresponding  to  special 
positions  of  the  faces  of  the  general  form.  These  may  be  tabu- 
lated as  follows  : 


Position  of  Any  Face  and  its  Pole. 

Symbol* 

Name  of   Form. 

Classes  to  which 
Form  Belongs. 

I.  Parallel  to  a  and  b  
Poles  are  projected  at  cen- 
ter. 

i°°'S 

BASAL  PINACOID,  Fig.  78. 
BASAL  PLANE. 
one  face  of  Fig  78. 

6,   -,      8. 

—  ,    7,     —  • 

2.   Parallel  to  a  and  c. 

PINACOID,  Fig.  79. 

6       7,      8 

Poles   are   at   intersections 
of  b  axis   and   primitive 
circle  ....            .    . 

<OI05 

(Brachy  Pinacoid). 

3.  Parallel  to  b  and  c  .    .    .    . 
Poles  are  at  intersections  of 
a  axis  and  primitive  cir- 
cle     

§'°°! 

PINACOID,  Fig.  80. 
(Macro  Pinacoid). 

6,      7.      8. 

4.  Parallel  to  a.     Poles  are  on 
the   b  axis 

\*ki\ 

PRISM,  Fig.  81. 
(Brachy  Dome). 

6,    -,      8. 

5.  Parallel  to  b.     Poles  are  on 
the  a  axis  .... 

{*,/] 

DOME,  Fig.  84. 

PRISM,  Fig.  82. 
(Macro  Dome). 

—  ,      7,    —  • 
6,    -,      8. 

6.   Parallel  to  c.     Poles   are   on 
the  primitive   circle  .    .    . 

\Uv] 

DOME,  Fig.  85. 

PRISM,  Fig.  83. 
(Rhombic  Prism). 

—      7»   — 
6,      7,      8. 

FIG.  8 1.  FIG    82.  FIG.  83. 

LIMIT  FORMS  OF  CLASS  8. 

*  To  obtain  type  symbols.      The  order  is  invariably  hkl  with  reference  to  a,  bt 
Any  may  become  zero. 


CHARACTERS  OF  CRYSTALS. 


FIG.  84.  FIG.  85. 

OTHER  LIMIT  FORMS  WHICH  ARE  NEW  SHAPES. 

PROJECTION  AND  CALCULATION  OF  ORTHORHOMBIC  FORMS. 
The  pinacoid  (ooi)  is  usually  selected  as  the  plane  of  projection. 
The  poles  are  as  in  the  table  and  their  exact  positions  usually  re- 
sult from  the  intersections  of  known  zones  or  by  Problem  3,  p.  22. 
It  is  sometimes  convenient  to  calculate  the  position  of  a  plane 
(hko]  corresponding  to  the  plane  (hkl),  lay  off  this  on  the  primitive 
circle  thus  determining  the  zone  [/z/£o  ooi]. 

CALCULATION  OF  ELEMENTS. 

The  interaxial  angles  are  all  right  angles. 

In  the  spherical  triangle  rst,  t  =  90°,  r  =  hkl :  oio,  s  =  hkl\  100 
Substituting  in  formulae  p.  34. 


cos  s  cos  r 

cos  tr  =  -7 ,  cos  ts  =  - — 

sin  r  sin  s 


OL  OL 

m~cot&,m 


cot 


a  =  OH.k,  b=OK.k,  c  =  OL.l. 

Also,       a  =  tan  y2  (100  :  1 10)  and  c  =  tan  y2  (01 1  :  01 1) 
=  a  tan  y2  (101  :Toi). 


Vihol 


A 100 


FIG.  86. 


FIG.  87. 


GEOMETRICAL   CHARACTERS. 


GENERAL  EQUATION  BETWEEN  AXES  AND  INDICES. 

y  cos  PA  =  t  cos  PB,  =  4  cos  ^ 

h  k  I 

To  DETERMINE  *  ANY  POLE  P. 

cot  PA  =  -^  cos  /**=  £  cos 


cot  P£  =  ^  cos  PBC=~cvs  PBA  =  ^— = 


kc 
tan  PC  A  =  4/  ,  tan  PBC=~,  tan 


For  Unit  Plane  O 
a:b\c=co$OB  cos  OC  :  cos  <9<7  cos   OA\  cos  O4  cos  OB. 

To  FIND  THE  ARC  JOINING  ANY  Two  POLES  Pand  P' 

Let  5  denote  JPPc*  +  /&W  +  /2^2,  5r  a  similar  quantity  from 
indices  h'k'l'  of  second  pole. 

IfP'  =  A,  B,  C,  H,  Kor  L  of  Fig.  87 


cos  P(7  =  sin  PZ  = 


If  P  and  P'  are  faces  of  the  same  form  f 

a     I 


tan  P£^  =  j  ^  ;  tan  J  (//^/)  :  (^/)  =  -  -  j  cos 


sin  J  (//>&/)  :  (//J/)  =  cos  \  (hkl)  :  (hkT)  •  cos  PC  A 
sin  J  (tiki)  :  Qikl)  =  cos  J  (//>&/)  :  (^7)  sin  PC4. 

TANGENT  RELATION  BETWEEN  ^/=  P,  and  h'k'l—  P1  which  lie  in 

a  zone  with  A,  B  or  (7. 

^     tanPM_^/_//>  k'tenP'B  _lr  _hr  m  I'     tenP'C_hf  _k' 
H  '  tan  PA~~k~~T'  ~k    tanP£  ~7~h  '  7  "  tan/^~~/£~J 

*  Miller's  Crystallo'graphy,  p.  79,  and  Story-Maskelyne's  Crystallography,  p.  443. 
fGroth,  jP^/j.  Kryst.,  p.  574. 


CHARACTERS  OF  CRYSTALS. 


TETRAGONAL  SYSTEM. 


FJG.  88. 


All  forms  of  this  system  must 
be  referable  to  two  equivalent 
axes,  a,  at  90°  to  each  other, 
Fig.  88,  and  the  axis  c,  conven- 
tionally vertical,  at  90°  to  both. 

The  forms  possible  on  crystals 
of  the  same  substance  can  all  be 


referred  to  one  value  of  - 
a 


9.  CLASS  OF  THE  THIRD  ORDER  BISPHENOID. 
With  composite  symmetry  to  a  quaternary  axis  and  a  plane  at 
right  angles  thereto.  As  shown  in  the  projection,  Fig.  89,  four 
faces  satisfy  the  symmetry  for  the  most  general  form  or  TETRA- 
GONAL BISPHENOID  OF  THIRD  ORDER,  Fig.  90.  No  examples  are 
known. 


\ 

O      : 

—fa 


a 

FJG.  89. 


FIG.  90. 


10.  CLASS  OF  THE  TETRAGONAL  PYRAMID  OF  THIRD  ORDER. 
With  one  axis  of  quaternary  symmetry.     As  shown  in  the  stere- 
ographic  projection,  Fig.  91,  four  faces  satisfy  the  symmetry  for 
the  most  general  form  or  TETRAGONAL  PYRAMID  OF  THIRD  ORDER, 
Fig.  92.     EXAMPLE. — Wulfenite. 


a 

FIG.  91. 


FIG.  92. 


GEOMETRICAL   CHARACTERS.  41 

II.    SCALENOHEDRAL  CLASS. 

With  two  planes  of  symmetry  at  right  angles  to  each  other  and 
intersecting  in  an  axis  of  quaternary  symmetry.  Also  two  axes 
of  binary  symmetry  midway  between  the  planes.  As  shown  in 
the  projection,  Fig.  93,  eight  faces  satisfy  the  symmetry  for  the 
most  general  form  or  SCALENOHEDRON,  Fig.  94.  EXAMPLE. — Chal- 
copyrite. 


FIG.  93. 


FIG.  94. 


12.  TRAPEZOHEDRAL.  CLASS. 

Without  planes  of  symmetry,  but  with  the  five  axes  of  Class  15. 
As  shown  in  the  projection,  Fig.  95,  eight  faces  satisfy  the  sym- 
metry for  the  most  general  form  or  TRAPEZOHEDRON,  Fig.  96. 
EXAMPLE.— Nickel  sulphate,  NiSO4.6H2O. 


/„     \ 


r 


KO 


FIG.  95. 


FIG.  96. 


13.  CLASS  OF  THE  TETRAGONAL  BIPYRAMID  OF  THIRD  ORDER. 
With  one  horizontal  plane  of  symmetry  and  one  vertical  axis  of 
quaternary  symmetry.     As  shown  in  the  projection,  Fig.  97,  eight 
faces   satisfy  the  symmetry  for  the  most  general  form  or  TETRA- 
GONAL BIPYRAMID  OF  THIRD  ORDER,  Fig.  98.    EXAMPLE. — Scheelite. 


CHARACTERS  OF  CRYSTALS. 


FIG.  98. 


14.  CLASS  OF  THE  DlTETRAGONAL  PYRAMID. 

With  four  vertical  planes  of  symmetry  intersecting  in  an  axis  of 
quaternary  symmetry.  As  shown  in  the  projection,  Fig.  99,  the 
essential  change  from  Class  15  is  the  omission  of  the  plane  of  sym- 
metry normal  to  the  quaternary  axis.  The  general  form  is  there- 
fore geometrically  like  the  upper  or  lower  half  of  Fig.  103.  Fig. 
100  shows  the  six- faced  most  general  form  or  DITETRAGONAL 
PYRAMID.  EXAMPLE. — Silver  fluoride,  AgF.H2O. 


FIG.  ico. 


15.  CLASS  OF  THE  DITETRAGONAL  BIPYRAMID. 

With  four  planes  of  symmetry  at  45°  to  each  other,  which  inter- 
sect in  an  axis  of  quaternary  symmetry,  and  one  plane  normal  to 
these  which  intersects  the  four  planes  in  axes  of  binary  symmetry. 

The  planes  divide  space  into  sixteen  sections,  Fig.  101  shown 
in  the  projection,  Fig.  102,  as  eight  birectangular  spherical  tri- 
angles with  angles  at  the  center  of  45°.  Any  upper  face  corre- 
sponding to  a  pole  X,  by  rotations  of  90°  coincides  successively 
with  three  other  upper  faces.  These  four  reflected  in  a  vertical 
planes  coincide  with  four  others  and  the  eight  reflected  in  a  hori- 
zontal plane  coincide  with  eight  lower  faces  marked  with  a  circle. 


GEOMETRICAL  CHARACTERS. 


43 


FIG.  10 1. 


FIG.  103. 


Since  /remains  with  the  third  axis,  the  letter  permutations  are 
only  /^/and  khl.  Each  of  these  is  subject  to  eight  permutations  in 
sign  +  +  +  ,+»  +  ,  ---  +  ,-  +  +,+  +  -,+  --  ,  ---  ,-  +  —  . 

Both  symmetry  and  permutations  show  that  there  must  be  in 
this  class  sixteen  faces  in  the  most  general  form  or  DITETRAGONAL 
BIPYRAMID  Fig.  103.  EXAMPLES.  —  Zircon,  cassiterite,  rutile. 


LIMIT  FORMS  OF  CLASS  15. 


FIG.  104. 


FIG.  105. 


FIG.  106. 


FIG.  107. 


FIG.  108. 


FIG.  109. 


44 


CHARACTERS  OF  CRYSTALS. 


THE  Six  LIMIT  FORMS. 

In  each  class  there  are  six  limit  forms  corresponding  to  special 
positions  of  the  faces  of  the  general  form.  These  may  be  tabulated 
as  follows: 


Position  of  any  Face  and  its 
Pole. 

Symbol* 

Name  of  Form. 

Classes  to  which  the  Form 
belongs. 

I.  Intersects  the    vertical 

5OOI  i 

* 

BASAL  PINACOID,  Fig. 

15,—,  13,12,  II,—  ,9. 

axis    and   is  parallel  to 

104. 

both  basal  axes.     Poles 

are    projected    at     the 

BASAL  PLANE,  one  face 

—,14,—,—,—,  io,—. 

center. 

of  Fig.  104. 

2.  Intersects  the    vertical 

\hol\ 

TETRAGONAL  BIPYRA- 

15,—  ,13,   12,   II,—,—. 

axis  and  is  parallel  to 

MID,  SECOND  ORDER, 

one   basal   axis.     Poles 

Fig.  105. 

tire  on  cixicil  dicimctcrs 

TETRAGONAL  PYRAMID 

IA    —           —  '  io   

SECOND  ORDER,  Fig. 

HTc'T'D  A  /TM\T  A  T          T^TCT>TTT7 

Q 

J.  ii.1  KALrUWAL*      X51bl  rill*- 

NOID,    SECOND    OR- 

DER, Fig.  in. 

3.    Intersects  the  vertical 

\hhl\ 

TETRAGONAL    BIPYRA- 

15,  —  ,  13,  12,  —  ,  —  ,  —  . 

axis  and  is  equally  in- 

i        5 

MID,    FIRST  ORDER, 

clined     to    both     basal 

Fig.  106. 

axes.     Poles  are  on  di- 

TETRAGONAL PYRAMID, 

—  ,  14,—,—  ,  —  ,  io,—. 

agonal  diameters. 

FIRST    ORDER,    Fig. 

114. 

TETRAGONAL   BISPHE- 

NOID,  FIRST  ORDER, 

Fig.  no. 

4.  Parallel  to  the  vertical 
axis  and   to   one   basal 
axis.    Poles  are  at  inter- 

5.00J 

TETRAGONAL     PRISM, 
SECOND  ORDER,  Fig. 

15,14,13,12,  ii,  io,  9. 

sections  of  primitive  cir- 

107. 

cle  and  axial  diameters. 

5.  Parallel  to  the  vertical 
axis  and  equally  inclined 
to  the  basal  axes.   Poles 

{110} 

TETRAGONAL     PRISM, 
FiRST    ORDER,    Fig. 

15,  14,  13,  12,  II,  10,  9. 

are   at    intersections   of 

1     • 

primitive  circle  and  di- 

agonal diameters. 

6.  Parallel  to  the  vertical 

\  hkv  \ 

DlTETRAGONAL  PRISM, 

15,  14,—  ,12,  II,—,  —  .- 

axis   and  unequally  in- 

<       > 

Fig.  109. 

clined  to  the  basal  axes. 

TETRAGONAL     PRISM, 

—  ,  —  ,  13,  —  ,  —  ,  io,  9. 

Poles  are  on  the  primi- 

THIRD ORDER,  Fig. 

tive  circle. 

112. 

*  The  first  and  second  indices  will  be  h  and  k  or  both  h  if  equal,  or  ho  if  the  face 
is  parallel  to  one  basal  axis.  The  third  symbol  will  be  i  or  o  as  the  face  intersects  or 
is  parallel  to  the  vertical  axis.  These  must  be  reduced  to  simplest  form. 


GEOMETRICAL   CHARACTERS. 
LIMIT  FORMS  OF  CLASS  15. 


45 


FIG.  112. 

OTHER  LIMIT 

FORMS  WHICH 

ARE  NEW 

SHAPES. 


FIG.  113.  FIG.  114. 

PROJECTION  AND. CALCULATION  OF  TETRAGONAL  FORMS. 
The  basal  pinacoid  is  usually  selected  as  the  plane  of  projec- 
tion, the  poles  lying  as  stated  in  the  table.     On  p.  23,  Fig.  36,  is 
described  the  projection  of  the  poles  of  a  crystal  of  cassiterite. 

DETERMINATION  OF  ELEMENTS. 

In  Fig.  86,  cos  tr  =  -.    -  and  — —  =  cot  tr.    c  =  OL./ 
sin  r         OH 

Also  in  Fig.  115  if  hhl  =  F. 

c  =  tan  FC  cos  45°  =  tan  KC 
sin  KC  =  tan  KF  cot  45 

tan  PC 


To  DETERMINE  POSITION  OF  ANY 
POLE*  P. 

cot  PA  =  tan  PH  =  {*  cos  PAB 


=  k,C  cos  PAC  = 
la 


ch 


Lk&o 


Aioo 


Miller's  Crystallography,  p.  44,  Story-Maskelyne's  Crystallography,  p.  449. 


46 


CHARACTERS  OF  CRYSTALS. 


cot  PB  =  tan  PK  =  f-  cos  PBA  =  ^cos  PBC 

la 


cot  PC  =  tan  PL  =  ™  cos  PCA  =  =-  cos  PCB  = 
kc  kc 

GENERAL  EQUATION  BETWEEN  AXES  AND  INDICES. 


-cosPA  =  ~ 
n,  k 


=     cos  PC. 

/ 


FORMULA  FOR  ANGLE  BETWEEN  Two  PLANES. 
cos  PP  =  . 


la 


Formulae  for  angle  between  two  faces  of  same  form  are  as  on 
P-  39- 


ANGLE  PC  WHEN  PC  KNOWN 


tan  PC 


tan  PC. 


HEXAGONAL  SYSTEM. 

All  forms  in  this  system  must  be  referable  to  three  equivalent 
axes  a  in  one  plane  at  60°  to  each  other  and  a  fourth  axis  c  nor- 
mal to  these,  conventionally  placed  vertically.  Fig.  116. 

The  system  comprises  two  grand 
divisions.  The  Hexagonal  division 
with  five  classes  in  which  the  fourth 
axis  is  an  axis  of  senary  symmetry. 
The  Rhombohedral  division  with 
seven  classes  in  which  this  axis  is 
2  an  axis  of  ternary  symmetry. 

The  vertical  and  basal  axes  being 
non-equivalent,   the    ratio    of    the 

parameters  -  is  always  an  irrational 

number  and  for  each  different  ratio  a  different  series  of  forms 
exists,  one  series  only  being  capable  of  occurrence  on  the  crystals 
of  the  same  substance. 

The  basal  axes  are  most  conveniently  considered  in  the  order  of 


GEOMETRICAL  CHARACTERS. 


47 


the  figure,  for  then*  the  third  index  is  always  the  algebraic  sum  of 
the  first  and  second. 


THE  RHOMBOHEDRAL  DIVISION. 
1 6.  CLASS  OF  THE  TRIGONAL  PYRAMID  OF  THIRD  ORDER. 
Without  planes  of  symmetry,  but  with  an  axis  of  ternary  sym- 
metry.    As  shown  in  projection,  Fig.  117,  the  symmetry  is  satis- 
fied by  three  faces  for  the  general  form,  or  TRIGONAL  PYRAMID 
OF  THIRD    ORDER,   Fig.    118.       EXAMPLES. — Sodium    periodate 
NaIO4.sH2O. 


V: y 

FIG.  117. 


FIG.  1 1 8. 


17.  CLASS  OF  RHOMBOHEDRON  OF  THE  THIRD  ORDER. 
With  composite  symmetry  to  a  ternary  axis  and  a  plane  at  right 
angles  thereto.  As  shown  in  the  stereographic  projection,  Fig. 
119,  the  symmetry  is  satisfied  by  six  faces  for  the  most  general 
form  or  RHOMBOHEDRON  OF  THE  THIRD  ORDER,  Fig.  120.  EX- 
AMPLES.— Dioptase,  phenacite. 


o   • 
ia 


FIG.  120. 


FIG.  119. 

1 8.  CLASS  OF  THE  TRIGONAL  TRAPEZOHEDRON. 
Without  planes  of  symmetry,  but  with  the  four  axes  of  sym- 
metry of  Class  22.     As  shown  in  projection,  Fig.  1 21,  the  sym- 
metry is  satisfied  by  six  planes  for  the  general  form,  or  TRIGONAL 
TRAPEZOHEDRON,  Fig.  122.     EXAMPLES. — Quartz,  cinnabar. 

*  Simple  proof  in  Bauerman's  Systematic  Mineralogy,  p.  76. 


48 


CHARACTERS  OF  CRYSTALS. 


FIG.  1.2 1.  FIG.  122. 

19.  CLASS  OF  THE  TRIGONAL  BIPYRAMID  OF  THIRD  ORDER. 
With  one  plane  of  symmetry  at  right  angles  to  the  axis  of  ter- 
nary symmetry.  As  shown  in  the  projection,  Fig.  123,  the  sym- 
metry is  satisfied  by  six  planes  for  the  most  general  form,  or  TRI- 
GONAL BIPYRAMID  OF  THIRD  ORDER,  Fig.  124.  No  examples  are 
known. 


FIG.  1.23.  FIG.  124. 

2O.    CLASS  OF  THE  DlTRIGONAL  PYRAMID. 

With  three  planes  of  symmetry  at  60°  to  each  other  which  in- 
tersect in  the  axis  of  ternary  symmetry.  These  forms  are  geomet- 
rically the  upper  or  lower  halves  of  those  of  Class  22.  As  shown 
in  projection,  Fig.  125,  the  symmetry  is  satisfied  by  six  planes 
for  the  most  general  form,  or  DITRIGONAL  PYRAMID,  Fig.  126. 
EXAMPLES. — Tourmaline,  proustite,  pyrargyrite. 


FIG.  125. 


FIG.  126. 


21.  CLASS  OF  THE  DITRIGONAL  SCALENOHEDRON. 
With  three  planes  of  symmetry  at  60°  to  each  other  and  inter- 
secting in  an  axis  of  ternary  symmetry,  and  three  axes  of  binary 


GEOMETRICAL   CHARACTERS. 


49 


symmetry  bisecting  the  angles  between  the  planes.  As  shown  in 
projection,  Fig.  127,  the  symmetry  is  satisfied  by  twelve  faces  for 
the  most  general  form,  or  DITRIGONAL  SCALENOHEDRON,  Fig.  128. 
EXAMPLES. — Calcite,  hematite,  corundum. 


FIG.  127. 


FIG.  128. 


22.  CLASS  OF  THE  DITRIGONAL  BIPYRAMID. 

With  three  planes  of  symmetry  at  60°  to  each  other  which  in- 
tersect in  an  axis  of  ternary  symmetry  and  one  plane  normal  to 
these  which  intersects  the  others  in  axes  of  binary  symmetry. 
The  planes,  Fig.  129,  divide  space  into  twelve  equal  parts,  shown 
in  projection,  Fig.  130,  as  six  birectangular  spherical  triangles 
with  angles  at  the  center  60°. 

Anyjapper  face,  corresponding  to  a  pole  X  in  one  of  the  tri- 
angles, would  by  rotations  of  120°  around  the  ternary  axis  coin-, 
cide  with  two  other  upper  faces.  Each  of  these  by  reflection  in 
a  vertical  plane  of  symmetry  coincides  with  another  face  and 
the  sixjby  reflection  in  the  horizontal  plane  of  symmetry  coincide 
with  six  lower  faces,  the  poles  of  which  are  indicated  by  circles. 

The  symmetry  of  the  class  is  satisfied  by  twelve  faces  for  the 
most  general  form  or  DITRIGONAL  BIPYRAMID,  Fig.  131.  No  ex- 
amples are  known. 


FIG.  129. 


FIG.  130. 


CHARACTERS  OF  CRYSTALS. 


THE  Six  LIMIT  FORMS. 

In  each  class  there  are  six  limit  forms  corresponding  to  special 
positions  of  the  faces  of  the  general  form.  These  may  be  tabu- 
lated as  follows : 


Position  of  any  Face  and  its 
Pole. 

Symbol* 

Name  of  Form. 

Classes  to  which  the  forrn 
belongs. 

I.  Parallel  to  all  basal 
axes.  Poles  projected  at 
the  centre. 

2.  Oblique  to  the  vertical 
axis  but  parallel  to  one 
basal  axis.  Poles  are  on 

joooij 
ShoTiii 

BASAL  PINACOID,  Fig. 
153- 
BASAL  PLANE,one  plane 
of  Fig.  153. 
TRIGONAL  BIPYRAMID 
FIRST    ORDER,    Fig. 
132. 
RHOMBOHEDRON  FIRST 

22,21,—  ,19,  18,17,—  . 
—,—,20,—,—,—,  1  6. 
22,—  ,—  ,  19,—  ,—  ,—  . 

the  interaxial  diameters. 

ORDER,  Fig.  135. 
TRIGONAL  PYRAMID 

3.  Oblique  to  the  vertical 
axis  and  equally  inclined 
to  alternate  basal  axes. 

Shfahil 

FIRST    ORDER,    Fig. 
136. 
HEXAGONAL    BIPYRA- 
MID SECOND  ORDER, 

Fig.  155- 
HEXAGONAL  PYRAMID 

22,21,—  ,—,—,—,—. 

Poles  are  on  the  axes. 

SECOND  ORDER,  Fig. 
1  60. 
TRIGONAL  BIPYRAMID 
SECOND  ORDER,  Fig. 

—,—,—,19,18,—,—. 

17       -' 

OND  ORDER,  Fig.  140. 
TRIGONAL      PYRAMID 

>        >        >        >       »*/» 

,16. 

4.  Parallel  to  the  vertical 
axis  and   to   one   basal 
axis.     Poles  are  at  inter- 
sections of  primitive  cir- 
cle   and    interaxial   di- 
ameters. 
5.  Parallel  to  the  vertical 
axis  and  equally  inclined 
to  alternate  basal  axes. 
Poles  are  at  intersections 
of   axes    and   primitive 
circle. 

6.  Parallel  to  the  vertical 
and  unequally  inclined  to 
all  basal  axes.  Poles  are 

S  loloj 

{"*} 

SECOND  ORDER,  Fig. 
141. 
TRIGONAL  PRISM  FIRST 
ORDER,  Fig.  133. 
HEXAGONAL     PRISM 
FIRST    ORDER,    Fig. 
156. 

HEXAGONAL    PRISM 
SECOND  ORDER,  Fig. 

"57- 

TRIGONAL  PRISM  SEC- 
OND ORDER,  Fig.  138. 

DITRIGONAL   PRISM, 
Fig.  134- 

DlHEXAGONAL    PRISM, 

22,  21,20,  19,  —  ,  —  ,  16. 
—,—,—,—,18,17,—. 

22,21,20,       -,l8,I7,—. 

—  ,—  ,—  ,19,—  ,—  ,16. 

22,—  ,20,—  ,8  1,—,—. 
,21,       ,       ,      ,       , 

Fig.  158. 
TRIGONAL       PRISM 
THIRD  ORDER,  Fig. 

139. 
HEXAGONAL      PRISM 
THIRD  ORDER,  Fig. 
161. 

—  ,—  ,—  ,  19,—  ,—  ,  1  6. 

—,—,—,—  ,—,17,—  . 

*  See  footnote  under  hexagonal  division,  p.  54. 


GEOMETRICAL  CHARACTERS. 


FIG.  132. 


FIG.  133. 


FIG.  134. 


FIG.  135. 


FIG.  136. 


FIG.  137. 


FIG.  138. 


FIG.  139. 


FIG.  140. 


LIMIT  FORMS  IN  RHOMBOHEDRAL  DIVISION 

WHICH  ARE  GEOMETRICALLY  SHAPES 

NOT  INCLUDED  IN  HEXAGONAL 

DIVISION. 


FIG.  141. 


52  CHARACTERS  OF  CRYSTALS. 

THE  HEXAGONAL  DIVISION. 
23.  CLASS  OF  THE  THIRD  ORDER  HEXAGONAL  PYRAMID. 
Without  planes  of  symmetry  but  with  one  axis  of  senary  sym- 
metry.    As  shown  in  the  projection,  Fig.    142,  six  planes  satisfy 
the  symmetry  of  the  most  general  form  or  HEXAGONAL  PYRAMID 
OF  THIRD  ORDER,  Fig.  143.     EXAMPLE. — Nephelite. 


FIG.  142.  FIG.   143. 

24.  CLASS  OF  THE  HEXAGONAL  TRAPEZOHEDRON. 
Without  planes  of  symmetry  but  with  the  seven  axes  of  Class 
27.     As  shown  in  projection,  Fig.   144,  twelve  faces  satisfy  the 
symmetry  for  the  most  general  form  or  HEXAGONAL  TRAPEZOHE- 
DRON, Fig.  145.     EXAMPLE.— (SbO)2  BaC4H4O6.KNO3. 


\ 


°v 


FIG.  144.  FIG.  145. 

25.  CLASS  OF  THE  THIRD  ORDER  HEXAGONAL  BIPYRAMID. 
With  one  plane  of  symmetry  normal  to  the  axis  of  senary  sym- 
metry.    As  shown  in  projection,  Fig.  146,  twelve  faces  satisfy  the 
symmetry  for  the  most  general  form  or  HEXAGONAL  BIPYRAMID  OF 
THIRD  ORDER,  Fig.  147.     EXAMPLE. — Apatite,  pyromorphite. 


FIG.  146. 


FIG.  147. 


GEOMETRICAL   CHARACTERS. 


53 


26.  CLASS  OF  THE  DlHEXAGONAL  PYRAMID. 

With  six  planes  of  symmetry  at  30°  to  each  other  and  intersec- 
ting in  the  axis  of  senary  symmetry.  From  the  projection,  Fig. 
148,  it  is  evident  that  the  general  form,  or  DIHEXAGONAL  PYRAMID, 
Fig.  149,  is  geometrically  the  upper  (or  lower)  half  of  the  general 
form,  Fig.  152,  of  the  next  class. 


FIG.  148. 


FIG.  149. 


27.    CLASS    OF   THE    DIHEXAGONAL   BlPYRAMID. 

With  seven  planes  of  symmetry,  six  being  normal  to  the  seventh 
and  at  30°  to  each  other  as  shown  in  Fig.  150.  The  common 
intersection  of  six  planes  of  symmetry  is  the  axis  of  senary  sym- 
metry and  their  intersections  with  the  seventh  are  axes  of  binary 
symmetry. 


FIG.   150. 


FIG.    152. 


In  projection,  Fig.  151,  the  planes  of  symmetry  form  twelve 
equal  birectangular  spherical  triangles  with  angles  at  the  centre, 
30°.  Any  upper  face,  with  a  pole  marked  x ,  by  rotations  of  60° 
around  the  senary  axis  coincides  successively  with  five  other  upper 
faces  represented  by  crosses  in  the  alternate  triangles.  Each 
of  the  six  reflected  in  a  vertical  plane  of  symmetry  coincides 
with  another  face  and  the  twelve  reflected  in  the  horizontal 
plane  coincide  with  twelve  lower  faces  represented  by  circles. 


54 


CHARACTERS  OF  CRYSTALS. 


The  letter  permutations  of  the  general  symbol  (1ikli\  in  which  i 
always  must  remain  in  fourth  place,  are  hkli,  hlki,  khli,  klhi,  Ihki, 

Ikhi,  and  each  may  have  four  sign  permutations,  +-j f-, j-+, 

+  -j and 1 —  or  twenty-four  permutations  in  all.  Pro- 
jection and  permutation  both  show  that  the  symmetry  of  the  class 
is  satisfied  by  twenty-four  faces  for  the  most  general  form  or  Di- 

HEXAGONAL  BlPYRAMID,  Fig.   152.      EXAMPLE. Beryl. 


THE  Six  LIMIT  FORMS. 

In  each  class  there  are  six  limit  forms  corresponding  to  special 
positions  of  the  faces  of  the  general  form.  These  may  be  tabu- 
lated a  follows : 


Position  of  any  Face  and  its  Pole. 


Symbol* 


Name  of  Form. 


Classes  to  which 
the  formbel'ng's 


I.  Parallel   to   all   basal   axes. 
Poles  projected  at  the  center. 


2.  Oblique  to  the  vertical  axis 
but  parallel  to  one  basal  axis. 
Poles  are  on  the  interaxial 
diameters. 

3.  Oblique  to  the  vertical  axis 
and  equally  inclined  to  alter- 
nate basal  axes.     Poles  are 
on  the  axes. 

4.  Parallel  to  the  vertical  axis 
and  to  one  basal  axis.    Poles 
are  at  intersections  of  primi- 
tive circle  and  interaxial  di- 
ameters. 

5.  Parallel  to  the  vertical  axis 
and  equally  inclined  to  alter- 
nate basal  axes.     Poles  are 
at  the  intersections  of  axes 
and  primitive  circle. 

6.  Parallel  to  the  vertical  but 
unequally     inclined    to    all  j 
basal  axes.     Poles  are  on  the 
primitive  circle. 


!OOOI 


\hhzhi\ 


1010 


1 1 20; 


BASAL  PINACOID,  Fig.  153. 
BASAL  PLANE,  one  plane  of 
Fig-  153- 


27,— ,25,24,— . 
—,26,— ,—,23. 


HEXAGONAL     BIPYRAMID  [27, — ,25,24, — . 

FIRST  ORDER,  Fig.  154.   | 
HEXAGONAL        PYRAMID  — ,26, — , — ,23. 

FIRST  ORDER,  Fig.  159. 

HEXAGONAL     BIPYRAMID  27, — ,25,24,- 

SECOND  ORDER,  Fig.  155. 
HEXAGONAL  PYRAMID  SEC- — ,26, — , — ,23. 

OND  ORDER,  Fig.  160. 

HEXAGONAL  PRISM  FIRST  127,26,25,24,23. 
ORDER,  Fig.  156. 


HEXAGONAL   PRISM    SEC- 
OND ORDER,  Fig.  157. 


27,26,25,24,23- 


DlHEXAGONAL  PRISM,  Fig.    27,26, — ,24, — . 

158. 

HEXAGONAL  PRISM  THIRD  — , — ,25,— ,23. 
ORDER,  Fig.  161. 


*  In  the  type  symbols  the  first  and  second  indices  relate  to  the  relative  inclination  of 
the  faces  to  the  alternate  basal  axes.  If  unequally  inclined  hk,  if  equally  inclined  hh, 
if  parallel  to  one  ho.  The  third  index  is  /,  the  algebraic  sum  of  the  first  and  second 
with  the  opposite  sign. 


GEOMETRICAL   CHARACTERS. 


55 


FIG.  153. 


FIG.  154. 


FIG.  155. 


FIG.  156. 


FIG.  157. 
LIMIT  FORMS  OF  CLASS  27. 


FIG.  158. 


FIG.  159.  FIG.  1 60. 

LIMIT  FORMS  GEOMETRICALLY  NEW  IN  THE 
OTHER  CLASSES. 

FIG.  161. 

PROJECTION  AND  CALCULATION   OF  HEXAGONAL  AND   RHOMBOHE- 

DRAL  FORMS. 

The  basal  pinacoid  is  the  plane  of  projection.     The  position  of 
the  poles  are  as  in  the  tables,  pp.  50  and  54. 

The  indices  of  a  face  truncating  an  edge  of  any  form  are  the 
sum  of  the  indices  of  the  faces  forming  the  edge,  for  example,  the 
edge  between  ion  and  oui  is  truncated  by  1122. 
THE  ZONAL  RELATIONS 

Are  calculated  from  three  indices  of  which  one  must  relate  to 
the  vertical  axis.     For  instance,  by  method  of  cross  multiplication 


CHARACTERS  OF  CRYSTALS. 


the  zone  indices  of  the  planes  hkliji'k'l'i' w\\\  be  determined  by  hki 
and  h'k'i' :  u  =  ki'  —  ik' ,  v  =  ih'  —  hi',  w  =  hk'  —  kh' . 

DETERMINATION  OF  ELEMENTS. 

Two  equal  basal  axes  at  120°  to  each  other  and  at  right  angles 
to  the  vertical  are  sufficient.  In  Fig.  -f u  the  spherical  triangle  rst 
has  /==  120°. 

cos  s  -f-  V2  cos  r  OL 

cos  tr  =  — - — : -^= — ;  cot  tr  =  ()rr]  c  =  OL.i 

or 

//          c 
tan  (oooi)  :(II22)  =  ^  =   -  =  c 


Cooo/JfT 


ono 


FIG.  162.  FIG.  163. 

To  DETERMINE  THE  POSITION  OF  ANY  POLE  P. 

Using  notation  of  Fig.  163  and  denoting  by  M  the  quantity 


31 

c(2/i  -f  k) 
cos  PA  =  -  --~,  cos 


m 
To  FIND  THE  ARC  JOINING  TWO  POLES  P  AND  P. 

cos  PP'  =  - 
TANGENT  PRINCIPLE. 


c(h  +  2k)  c(k  -  Ji) 

-  -^—  ,  cos  /^  =  A^- 

/v/3 


MM' 

tan  PC  _  k  _  h 
tarTF^  ~"  V  ~  Ji' 


GEOMETRICAL   CHARACTERS. 


57 


THE  ISOMETRIC  SYSTEM. 

All  forms  in  this  system  must  be  referrable  to  three  equivalent 
axes  at  right  angles  to  each  other.  The  system  includes  five 
classes. 

28.  CLASS  OF  THE  TETARTOID. 

The  forms  have  seven  axes  of  symmetry  but  no  planes  of 
symmetry.  Three  of  the  axes  are  binary  normal  to  the  faces  of 
the  hexahedron  and  four  are  ternary  through  diagonally  opposite 
solid  angles  of  the  hexahedron. 

As  shown  in  the  projection  Fig.  164,  twelve  faces  in  each  alter- 
nate octant  satisfy  the  symmetry  for  the  most  general  form  or 
TETARTOID,  Fig.  165.  EXAMPLES — Sodic  chlorate,  baric  nitrate. 


.-"•;;*.• -.. 


a 
FIG.  164. 


FIG.  165. 


29.  CLASS  OF  THE  GYROID. 

Without  planes  of  symmetry  but  with  all  the  axes  of  symmetry 
of  class  32.  As  shown  in  the  projection  Fig.  166,  twenty-four 
faces,  three  in  each  octant,  satisfy  the  symmetry  for  the  most  gen- 
eral form*  or  GYROID,  Figs.  167  and  168.  EXAMPLES, — Halite, 
sylvite,  cuprite. 


FIG.  1 66. 


FIG.  167. 


FIG.  i 68. 


*The  right  form,  Fig.  168,  and  the  left  form,  Fig.  167,  are  enantiomorphic,  that  is, 
their  elements  are  equal  and  the  faces  of  the  one  are  parallel  but  oppositely  placed 
with  respect  to  those  of  the  other. 


CHARACTERS  OF  CRYSTALS. 


30.  CLASS  OF  THE  DIPLOID. 

The  forms  have  three  cubic  planes  of  symmetry,  the  intersec- 
tion of  these  are  three  axes  of  binary  symmetry  and  there  are  four 
diagonal  axes  of  ternary  symmetry. 

As  shown  by  the  projection,  Fig.  169,  the  symmetry  of  the 
class  is  satisfied  by  twenty-four  faces  for  the  most  general  form  or 
DIPLOID,  Figs.  170  and  171.  EXAMPLES— Pyrite,  smaltite,  cobal- 
tite. 


FIG.  169. 


FIG.  170. 


FIG.  171. 


31.  CLASS  OF  THE  HEXTETRAHEDRON. 

The  forms  *  have  the  six  dodecahedral  planes  of  symmetry  of 
Class  32  and  the  seven  axes  of  symmetry  formed  by  their  intersec- 
tion. Of  the  latter,  four  are  ternary  and  three  binary. 

As  shown  in  the  projection,  Fig.  172,  the  symmetry  of  this 
class  is  satisfied  by  twenty-four  faces  for  the  most  general  form  or 
HEXTETRAHEDRON,  Figs.  173  and  174.  EXAMPLES. — Sphalerite, 
tetrahedrite,  diamond. 


FIG.  172. 


FIG.  173. 


FIG.  174. 


32.  CLASS  OF  THE  HEXOCTAHEDRON. 

/\ 
The  forms  of  this  class  have  three  planes  of  symmetry  parallel 

to  the  faces  of  the  cube  and  six  planes  of  symmetry  through  diag- 

*  The  forms  of  this  class  exist  in  pairs  which  are  said  to  be  congruent,  that  is  either 
by  a  revolution  of  90°  about  an  axis  may  brought  into  the  position  of  the  other.  They 
are  distinguished  as  right  and  left  forms. 


GEOMETRICAL    CHARACTERS. 


59 


onally  opposite  edges  of  the  cube,  that  is  parallel  to  the  faces  of 
the  rhombic  dodecahedron  Fig.  181. 


FIG.  175. 


FIG.  176. 


FIG.  i7|| 


The  intersections  of  these  planes  are  axes  of  symmetry;  three 
are  quaternary  the  intersections  of  the  planes  parallel  the  hexahe- 
dron;  four  are  ternary,  each  the  intersections  of  three  diagonal 
planes ;  and  six  are  binary,  each  the  intersection  of  a  plane  of  each 
kind. 

These  planes  and  axes  of  symmetry  are  shown  in  stereographic 
projection,  Fig.  176. 

The  number  of  faces  in  the  most  general  form  may  be  deter- 
mined as  follows : — 

In  any  quadrant  the  pole  marked  I  of  a  face  in  the  upper  half 
of  the  crystal  by  rotations  of  120°  around  the  ternary  axis  must 
coincide  successively  with  faces  the  poles  of  which  are  2  and  3. 
These  three  by  reflection  in  the  planes  of  symmetry  must  coincide 
I  with  4,  2  with  5,  3  with  6.  Since  there  is  a  vertical  axis  of 
quaternary  symmetry  the  entire  quadrant  must  coincide  succes- 
sively pole  for  pole  with  the  second,  third  and  fourth  quadrants 
and  finally  the  resulting  24  faces  reflected  in  the  horizontal 
plane  of  symmetry  must  coincide  with  24  lower  faces.  That 
is,  the  form  must  consist  of  48  faces  with  a  pole  in  each  of  the 
right  spherical  triangles  made  by  the  planes  of  symmetry. 

This  result  could  also  be  reached  by  considering  the  possible 
permutations  of  letter  and  sign  in  the  symbol  of  the  general  form 
\hkl  \ .  The  letter  permutations  are  : 

hkl,  hlk,  klh,  khl,  Ikh,  Ihk. 

Each  of  these  can  undergo  eight  permutations  in  sign  according 
to  the  octant  in  which  the  plane  occurs. 

Upper  octants  +  +  +,_  +  +,-      .  +,  +  —  +, 
Lower  octants  +  -\ , ( , ,  H , 


6o 


CHARACTERS  OF  CRYSTALS. 


or  48  permutations  in  all.  That  is  to  satisfy  the  symmetry  the 
most  general  form  or  HEXOCTAHEDRON,  Fig.  177,  must  consist  of 
48  faces  six  in  each  octant.  EXAMPLES.  —  Garnet,  fluorite,  spinel. 


THE  Six  LIMIT  FORMS. 

In  each  class  there  are  six  limit  forms  corresponding  to  special 
positions  of  the  faces  of  the  general  form.  These  may  be  tabu- 
lated as  follows  : 


Position  of  any  Face. 

Symbol* 

Name  of  Form. 

Classes  to  which 
Form  belongs. 

I.  Equally  inclined  to  all  three 
axes.  Poles  on  ternary  axes. 

J'"J 

OCTAHEDRON,  Fig.  178. 
TETRAHEDRON,  Fig.  184. 

32,—  ,30,29,—  . 
—,31,—  ,—,28. 

2.  Equally  inclined  to  two 
axes,  more  nearly  parallel 
the  third.  Poles  are  on  the 
short  s'des  of  triangles. 

{Ml] 

TRIGONAL  TRISOCT'AHEDRON, 
Fig.  179- 
TETRAGONAL    TRISTETRAHE- 
DRON,  Fig.  185. 

32,—  ,30,  29,—  . 

—  ,3'»—  —  .28- 

3.  Equally  inclined  to  two 
axes  less  nearly  parallel  the 
third.  Poles  are  on  hypothe- 
nuse. 

{Mi} 

TETRAGONAL      TRISOCTAHE- 

DRON,  Fig.   1  80. 

TRIGONAL         TRISTETRAHE- 
DRON,  Fig.  1  86. 

32,—,  30,29,—, 
—.31,—  ,—,28. 

4.  Equally  inclined  to  two 
axes,parallel  thethird.  Poles 
at  vertices  cf  right  angles. 

S"°i 

RHOMBIC      DODECAHEDRON, 
Fig.  181. 

32,31,30,29,28. 

5.  Unequally  inclined  to  two 
axes  parallel  thethird.  Poles 
are  on  the  long  sides. 

w 

TETRAHEXAHEDRON,  Fig.  182. 

PENTAGONAL         DODECAHE[ 
DRON,  Fig.  187. 

32,31,—  ,29,—  . 

—  ,—  ,  30  —  »28. 

6.  Parallel  to  two  axes.  Poles 
are  on  the  quaternary  axes. 

l,oo] 

HEXAHEDRON,  183. 

32,31,30,29,28. 

FIG.  178. 


FIG.  179. 


FIG.  1 80. 


*  The  type  symbols  of  the  forms  are  easily  obtained  without  direct  reference  to  the 
intercepts  by  noting  parallelism  and  relative  nearness  to  parallelism  to  the  axes.  Zero 
being  the  symbol  of  parallelism  and  conventionally  h  >  k  >  /. 


GEOMETRICAL   CHARACTERS. 


61 


FIG.  181. 


FIG.  182. 
LIMIT  FORMS  OF  CLASS  32. 


FIG.  183. 


FIG.  184. 


FIG.  185. 


FIG.  1 86. 


LIMIT  FORMS   GEOMETRICALLY  NEW 
THE  OTHER  CLASSES. 


IN 


FIG.   187. 

PROJECTION  AND  CALCULATION*  OF  ISOMETRIC  FORMS. 
The  forms  are  usually  projected  on  a  cubic  face  and  the  poles 
are  as  stated  in  the  table,  the  positions  being  usually  found  by  zone 
intersections.  The  planes  of  symmetry  divide  space  into  forty- 
eight  similar  parts  each  shown  in  stereographic  projection  as  a 
right  triangle  in  which :  the  hypothenuse  (54°  44'),  is  the  angle 
between  a  binary  and  a  ternary  axis  ;  the  longer  side  (45°)  is  the 
half  angle  between  two  binary  axes;  the  shorter  side  (35°  16')  is 
the  half  angle  between  two  ternary  axes. 

CALCULATION  OF  ELEMENTS. 

The  parameters  are  all  equal  and  the  angles  between  the  axes 
are  right  angles. 

*  Millers'  Crystallography,  pp.  25;  Story-Maskelyne's  Crystallography,  p.  453. 


62 


CHARACTERS  OF  CRYSTALS. 


GENERAL  EXPRESSION  OF  RELATION  BETWEEN  ANGLES  AND  INDICES* 

cos  PA  _  cos  PB  _  cos  PC 

~k~        ~T~        ~T~ 

To  DETERMINE  POSITION  OF  ANY  POLE.    P=(hkl). 
Adopting  notation  of  Fig.  188  equations  become 


tan  BAP=    \  tan  = 


cos  PA  =  sin  PH=  -^=; 

k 


Vihol 


Lk&Q 


cos  PC =  smPL  =  -Wn  which 


A  ioo 


FIG.  188. 

To  FIND  THE  ARC  JOINING  P=  hkl  and  P*  =  h'k'l' 

cos/y-^  +  ^tf 


CHAPTER  V. 


MEASUREMENT  OF  CRYSTAL  ANGLES. 


The  instruments  used  in  measuring  the  angles  between  the  faces 
of  crystals  are  called  goniometers,  the  simplest  form  being  the  appli- 
cation goniometer  invented  for 
Delisle  by  Carangeot  and  con- 
sisting of  a  semicircular  grad- 
uated arc  and  two  arms  moving 
upon  a  pivot,  the  position  of 
which  may  be  changed  accord- 
ing to  the  size  and  position  of 
the  crystal.  In  the  older  instru- 
ments the  arms  are  fastened  to 
FlG  l89-  the  arc,  but  in  the  later  types,  as 

in  Fig.  189,  they  are  detached  from  the  arc  during  measurement 
and  replaced  for  the  reading.  In  use  the  arms  are  each  in  close 
contact  with  a  face  and  at  right  angles  to  the  edge  between  the 
,  faces.  They  cannot  be  relied  upon  closer  than  one  degree,  and 
are,  therefore,  practically  limited  to  the  identification  of  previously 
measured  angles. 

All  measurements  of  ac- 
curacy are  obtained  by  ro- 
tating the  crystal  around 
the  edge  between  the  faces 
and  determining,  by  the  aid 
of  a  ray  of  light  fixed  in 
direction,  the  angle  between 
the  position  in  which  the 
first  face  gives  a  reflection 
and  that  in  which  the  sec- 
ond face  does  the  same. 

In  Fig.  190  the  crystal 

is  so  adjusted  that  an  edge  1(     ,90> 

coincides  with  the  axis  of 


64 


CHARACTERS  OF  CRYSTALS. 


rotation  O.  The  fixed  direction  of  the  ray  of  light  is  CO 7 
that  'is,  the  incident  ray  is  CO  and  the  eye  or  telescope  is  at 
T.  Then  OA  will  be  the  direction  of  the  first  face  when  it  acts  as 
a  reflecting  surface,  and  OB  will  be  the  direction  of  the  second 
face  at  that  time,  and  only  when  rotated  to  the  respective  direc- 
tions, OA'  and  OBft  will  a  reflection  be  obtained  from  the  second 
face,  that  is,  after  a  rotation  measured  by  the  equal  arcs  A  A' ,  BB1 
or  NN't  the  normal  angle  between  the  faces. 

GONIOMETERS  WITH  HORIZONTAL  AXES. 

The  original  reflection  goniometer  is  that  of  Wollaston.*  In 
this  a  base  and  column  support  in  a  collar  a  hollow  axle  to  which 
there  is  attached  at  one  end  a  vertical  disc  with  a  graduated  rim 
and  at  the  other  end  a  handle.  Through  this  axle  passes  a  second 
axle,  with  at  one  end  the  crystal  holder  and  at  the  other  a  handle. 
These  two  axles  may  be  clamped  to  turn  together,  or,  by  the  inner 
axle,  the  crystal  may  be  rotated  without  the  graduated  circle. 

In  the  simplest  forms  of  this  goniometer  the  crystal  is  fastened 


FIG.  191. 

with  wax  to  a  thin,  flexible,  brass  plate,  and  this  is  fixed  in  a 
holder  which  has  several  simple  motions  by  means  of  which  the 

*,W.  H.  Wollaston,  Description  of  a  reflective  goniometer,  Phil.  Trans.,  1809,  p. 
253-25.9. 


GEOMETRICAL   CHARACTERS.  65 

faces  of  the  zone  to  be  measured  can  be  made  to  project  clear  of 
the  apparatus  and  the  edge  to  coincide  with  the  axis  of  rotation. 
The  signal  may  be  a  horizontal  window  bar  twenty  to  thirty  feet 
away,  or  better,  a  horizontal  slit  in  a  screen  before  an  artificial  light 
and  it  must  be  parallel  to  the  axis  of  rotation.  The  eye  is  brought 
almost  into  contact  with  the  crystal  and  there  watches  for  the  re- 
flection of  the  signal  as  the  faces  successively  move  into  position. 

To  secure  a  constant  line  of  sight  a  reference  mark  may  be  made 
below  the  crystal  parallel  to  the  signal,  or  better,  a  second  image 
of  the  signal  may  be  obtained  in  a  small  mirror,  the  plane  of  which 
is  parallel  to  the  axis  of  rotation,  or  still  better,  a  telescope  with 
cross  hairs  may  be  used.  In  each  case  the  rotation  is  continued 
until  the  image  of  the  signal  is  bisected  by  or  coincides  with  the 
reference  mark. 

Mallard's*  modification  of  the  Wollaston  goniometer,  shown 
in  Fig.  191  on  the  left,  differs  from  the  earlier  types  in  the  sub- 
stitution of  a  better  crystal  holder ;  the  crystal  is  supported  in  the 
manner  suggested  by  Groth,  that  is,  it  is  fixed  with  wax  on  a  small 
circular  disc,  d,  and,  by  turning  the  screws  v,  v,  and  v'f  v' ,  can  re- 
ceive two  movements  of  rotation  on  two  arcs  of  circles  perpendic- 
ular to  each  other  which  have  their  common  center  near  the  middle 
of  the  crystal,  A,  so  that  the  changes  of  orientation  of  the  crystal 
do  not  too  much  displace  its  center  of  gravity. 
/  For  the  proper  centering,  the  entire  system  which  holds  the 
crystal  is  attached  to  two  sliding  planes,  g  and  g* ,  which,  by  means 
of  the  screws,  u  and  «',  impart  to  the  crystal  two  movements  of 
translation  in  a  plane  perpendicular  to  the  axis  of  rotation.  There 
is  also  a  tangent  screw  for  fine  rotation  and  the  mirror,  M,  has  four 
motions  of  adjustment  parallel  to  the  axis  of  rotation.  Any  signal 
may  be  used,  but  preferably  the  collimator  and  artificial  light,  as 
shown  in  the  figure. 

The  collimator  is  a  cylinder,  C,  with  at  one  end  a  large  lens,  L, 
and  at  the  other,  which  is  the  exact  focal  plane  of  the  lens,  an  ad- 
justable plate,  Fig.  192,  pierced  with  signals,  /,/',/",  of  various 
forms  beneath  each  of  which  is  a  fine  reference  slit.  The  light 
from  an  Argand  gas  lamp,  R,  or  a  Welsbach  burner  passes  through 
the  central  signals,  emerges  as  parallel  rays  from  the  lens  and  is 
reflected  to  the  eye  at  the  same  time  from  the  crystal  face  and  the 
mirror. 

*  Er.  Mallard,  Annales  des  Mines,  Nov.-Dec.  1887. 


66 


CHARACTERS  OF  CRYSTALS. 


A  greater  degree  of  accuracy  would  be  secured  by  the  addition 
of  an  observation  telescope,  as  in  the  Mitscherlich*  modification 
of  the  Wollaston,  but  this  is  accompanied  by  loss  of  light  and 
usually  makes  a  dark  room  necessary,  whereas  the  Mallard-Wol- 
laston  gives  good  results  in  ordinary  light. 


H — h 


ffl. 


FIG.  192. 


Professor  Groth  describes-)-  a  simple,  inexpensive  Wollaston  with 
a  telescope  fixed  parallel  to  the  plane  of  the  graduated  circle  and 
centered  on  the  goniometer  axis.  The  crystal  holder  is  like  that 
of  the  Mallard  instrument,  but  the  inner  axle,  which  ordinarily 
rotates  the  crystal  independently,  is  replaced  by  a  screw  which 
moves  the  crystal  horizontally  in  the  direction  of  the  axis.  The 
signal  used  by  Professor  Groth  is  a  very  small  incandescent  light  at 
a  distance  of  about  thirty  feet  where  it  appears  like  a  luminous 
point. 

The  principal  objection  to  the  Wollaston  type  of  goniometer  is 
that  the  weight  of  the  crystal  tends  to  throw  it  out  of  adjustment. 

GONIOMETERS  WITH  VERTICAL  AXES. 

Babinet,  v.  Lang,  Miller,  Websky  and  others  have  gradually  de- 
veloped this  more  perfect  and  more  generally  applicable  variety  of 
goniometer,  and  there  is  little  doubt  that  for  simplicity  of  adjust- 
ments and  perfection  of  construction  the  instrument  known  as  the 
Fuess  Model  II.  at  present  excels  all  others.  It  is  shown  in  Fig. 
193  and  consists  of: 

a.  THE  STAND.  A  tripod  with  a  conically  bored  head-piece  sup- 
porting three  concentric  axles — the  outer  axle  carrying  the  vernier 
circle  and  telescope,  and  turned  by  the  latter ;  the  fine  adjustment 
is  by  the  tangent  screw,  F,  and  the  clamp  screw,  «, — the  middle  axle 
carrying  the  graduated  circle  and  turned  by  the  disc,  g>  or  more 

*Ueber  ein  Goniometer  Abhandl.  Berlin  Akad ,  1843,  189-197. 
\Physikalische  Krystallographie,  III.  Ed.,  p.  613. 


GEOMETRICAL   CHARACTERS. 


67 


FIG.  193. 
/ 

conveniently  by  £•';*  the  fine  adjustment  is  by  the  tangent  screw, 
G,  and  the  clamp  screw,  /?, — the  inner  axle  guiding  the  rod  of  the 
crystal  carrier  and  which  may  be  turned  separately  or  clamped  to 
the  others. 

The  circle  is  graduated  to  half  degrees.  The  vernier  circle  is 
protected  by  a  ring,  A",  with  glass  windows  over  the  verniers,  while 
between  the  tripod  and  the  vernier  circle  is  a  ring  with  two  arms 
each  with  a  magnifier  and  a  mirror,  s,  by  means  of  which  the  vernier 
can  be  read  to  half  minutes  and  estimated  to  quarter  minutes. 

a.  THE  CRYSTAL  CARRIER  is  like  that  described  in  the  Mallard- 
Wollaston,  the  crystal  being  attached  to  a  platef  by  wax,  and  this  to 

*  The  part  g'  is  omitted  in  many  instruments  and  the  clamping  of  the  inner  axle  is 
by  a  vertical  screw. 

f  Two  or  three  sizes  are  furnished.  For  very  small  crystals  platinum  wire  may  be 
used  with  a  cement  of  gelatine  and  acetic  acid.  By  bending  the  wire  the  different 
zones  may  be  adjusted. 


68  CHARACTERS  OF  CRYSTALS. 

two  cylinder  arcs,  which  are  moved  by  tangent  screws  around  inter- 
secting axes  at  right  angles  to  each  other  and  to  the  goniometer  axis. 
The  common  point  of  intersection  is  within  the  crystal,  so  that 
their  motions  tip  the  crystal  without  moving  it  much  out  of  center. 
The  arcs  rest  upon  sliding  parts,  which  are  moved  by  micrometer 
screws,  in  the  direction  of  the  axes  of  the  arcs. 

c.  THE  TELESCOPE  L.     The  observation  telescope  turns  with  the 
vernier  circle,  to  which   it  is  attached   by  a   pillar,  and   its  axis 
always  intersects  and  is  perpendicular  to  the  axis  of  the  gonio- 
meter.    The  vertical  cross  hair  is  parallel  to  the  axis.     Before  the 
infinite  distance  objective  swings  an  extra  lens  of  a  focal  length 
equal  to  the  distance  to  the  goniometer  axis.     When  in  use  this 
converts  the  telescope  into  a  weak  microscope.     There  are  four 
eye  pieces  magnifying  respectively  six  diameters,  three  diameters 
(generally  used),  two  diameters  (for  distorted  reflections)  and   a 
diminishing  combination  which  is  used  with  very  small  faces  to  re- 
duce the  signal  to  one  third  of  its  natural  size. 

d.  The  COLLIMATOR.     The  infinite  distance  collimator  is  fixed 
firmly  to  the  tripod   by  the  pillar  C  and  its  axis  and  the  telescope 
axis  are  in  the  same  plane  perpendicular  to  the  goniometer- axis. 
The  signals  are  either  separate  tubes  or  all  four  may  be  on  a  revolv- 
ing target  in  the  focal  plane  of  the  collimator  lens. 

The  Schrauf  signal  is  a  light  diagonal  cross  on  a  dark  back 
ground,  with,  at  the  center,  very  minute  cross  hairs  which  are  used 
in  the  adjustments  and  with  rare  very  perfect  faces. 

The  Websky  Signal  is  an  orifice  bounded  by  two  circular  discs 
of  equal  diameter,  the  distance  apart  of  which  is  regulated  by  a 
screw.  This  admits  more  light  than  a  narrow  cleft  though  still 
very  narrow  in  the  center  and  if  dilated  by  a  narrow  face  (see  p. 
23)  and  diminished  in  intensity  the  shape  may  still  be  divided 
symmetrically  with  some  accuracy. 

The  Slit  Signal  is  an  adjustable  vertical  cleft  especially  adapted 
to  broad  faces  or  for  measurement  of  indices  of  refraction. 

The  Pinhole  Signalis  a  round  J^mm.  opening  used  in  adjusting 
and  in  recognizing  slight  deviations  from  the  zone  or,  by  multiple 
reflection,  cracked  and  facetted  faces. 

After  the  instrument  is  in  adjustment*  once  for  all  the  process 
of  measurement  is  as  follows  : 

*The  complete  adjustments  are  : 
I.  Adjtistmeiit  of  the  vertical  hair. 


GEOMETRICAL   CHARACTERS.  69 

Several  of  the  smallest  and  most  brilliant  crystals  are  selected, 
carefully  cleaned  with  chamois  skin  and  thereafter  handled  only 
with  pincers  or  a  tapering  pencil  of  wax.  Sketches  are  made 
showing  striations,  flaws  and  other  peculiarities  as  revealed  by  the 
magnifier  and  later  by  the  micro-telescope.  Letters  are  assigned 
to  all  faces  and  thereafter  represent  them. 

The  most  prominent  zone  is  first  measured  and  is  designated  by 
the  letters  of  its  two  most  prominent  faces,  say  [A  C~\.  The  crystal 
is  fastened  with  a  conical  pencil  of  wax  to  the  plate  so  that  the 
zone  axis  is  approximately  vertical  and  one  face  is  parallel  to  one 
of  the  motions  of  translation. 

The  crystal  is  brought  into  view  by  the  extra  lens  and  raised  or 
lowered  by  the  mother  screw  k.  With  one  arc  at  right  angles  to  the 
telescope  axis  one  of  the  edges  of  the  zone  is  made  coincident  with 

A  needle  is  adjusted  in  the  axis  of  the  goniometer,  so  that  during  rotation  it  does  not 
appear  to  move. 

An  eyepiece  is  adjusted  so  that  its  cross  hairs  are  distinct  and  is  then  placed  in 
the  telescope  tube,  focussed  on  some  distant  object,  the  extra  lens  dropped  into  position 
and  the  eyepiece  turned  until  one  hair  is  parallel  to  the  needle,  then  the  cross  hairs  are 
moved  horizontally  until  this  hair  coincides  with  the  needle. 

Making  Telesccope  axis  normal  to  Goniometer  axis. 

A  plate  of  parallel  glass  is  fastened  to  the  holder  approximately  parallel  to  the 
goniometer  axis  and  perpendicular  to  the  telescope  axis  and  is  made  exactly  so  by  re- 
flecting light  upon  the  cross  hairs  from  another  plate  fastened  in  front  of  the  eyepiece 
at  about  45°  to  the  telescope  axis.  The  reflected  light  throws  an  image  of  the  cross 
hairs  upon  the  parallel  glass,  which  is  reflected  back  into  the  telescope. 

By  aid  of  the  arcs  the  images  reflected  from  opposite  sides  of  the  parallel  glass  are 
made  to  take  the  same  position  in  the  field,  that  is,  the  plate  is  made  exactly  parallel  to 
the  axis  of  the  goniometer,  and  by  raising  or  lowering  the  cross  wires  their  center  is 
brought  into  coincidence  with  its  image  as  seen  after  reflection.  The  telescope  axis 
is  then  normal  to  the  plate  and^therefore  to  the  axis  of  the  goniometer. 
Adjustment  of  Horizontal  hair. 

During  rotation  of  the  outer  axle  the  image  of  the  intersection  of  the  cross  wires 
should  move  across  the  field  in  coincidence  with  the  horizontal  wire,  if  not,  raise,  lower, 
or  turn  the  wire  as  needed. 

The  Collitnator  Adjustment. 

The  pin  hole  signal  is  first  inserted,  focussed,  and  its  center  made  to  coincide  with 
the  horizontal  wire  of  the  telescope  by  the  screw  which  fastens  the  pillar  C  to  the  tripod. 
The  vertical  slit  and  Websky's  signal  are  focussed  and  made  symmetrical  to  the  tele- 
scope cross  hairs  and  finally  the  cross  hairs  of  the  Schrauf  signal  are  brought  into  ap- 
parent coincidence  by  their  traversing  screws. 
the  Other  Eyepieces, 

Are  adjusted  by  means  of  Schrauf 's  signal.  All  eyepieces  and  signals  have  sliding 
collars  with  a  projecting  tooth  which  fits  into  a  notch  in  the  tube  of  the  telescope  or  col- 
limator.  The  collar  is  tightened  when  the  tube  is  in  final  adjustment  and  thereafter  is 
fixed.  (M.  Websky,  Zeit.  /.  Krystallographie,  IV.,  545.) 


70  CHARACTERS  OF  CRYSTALS. 

the  vertical  hair  by  means  of  the  corresponding  micrometer  screws ; 
then  with  the  other  arc  at  right  angles  to  the  telescope  axis  the  same 
adjustment  is  made  with  the  other  screws,  and  this  is  repeated  until 
during  a  rotation  the  edge  and  vertical  hair  appear  to  be  one,  and  on 
raising  the  extra  lens  the  image  of  the  signal  from  any  face  in  the 
zone  is  symmetrically  bisected  by  the  cross  hairs. 

Except  in  the  case  of  very  narrow  faces*  it  is  not  necessary  to 
recenter  on  each  edge,  but  is  sufficient  by  a  slight  motion  with  the 
centering  screws  to  make  the  vertical  hair  coincident  with  an  imag- 
inary central  axis  within  the  zone. 

With  the  telescope  at  some  convenient  angle  to  the  collimator 
(100  to  1 20  degrees)  and  with  only /?  undamped,  the  graduated 
circle  and  -crystal  are  turned  together  by  the  disc  g  until  the  re- 
flected signal  is  seen  through  the  telescope,  then  ft  is  tightened,  the 
signal  moved  by  the  tangent  screw  G  until  it  is  bisected  by  the 
vertical  cross  hair  and  the  vernier  is  read  and  recorded. 

The  screw  /9  is  again  loosened  and  the  rotation  continued  until 
the  signal  is  received  from  a  second  face,  and  this  is  centered  by 
G  and  /?  and  recorded  as  before  and  the  difference  between  the  two 
readings  is  the  normal  angle.  The  position  of  a  third  face  is  de- 
termined in  the  same  way  and  so  on  around  the  zone. 

At  least  three  measurements  of  any  angle  should  be  made  and 
preferably  with  different  portions  of  the  circle.  For  the  second 
measurement  when  the  signal  from  the  second  face  is  centered  the 
clamp  /  is  loosened  and  the  crystal  alone  is  turned  until  the  signal 
from  the  first  face  is  centered,  then  /  is  tightened  and  /5  loosened  and 
the  crystal  and  circle  turned  together  as  before.  This  is  repeated 
before  a  third  measurement. 

It  is  important  to  assign  a  quality  mark  to  each  image  of  the 
signal.  For  instance,  a  distinct  image  may  be  2,  a  poor  image  I, 
a  band  or  shimmer  o,  and  the  proportionate  value  of  any  measure- 
ment may  then  be  taken  as  the  sum  of  the  quality  marks  of  the 
readings.  These  may  be  recorded  in  tabular  shape,  as  in  the  fol- 
lowing example : 

*  See  page  23. 


GEOMETRICAL  CHARACTERS. 


CRYSTAL  i.     Zone  [m  m'"] 


|         ! 

Face. 

READINGS. 

Mean 

Times  to  be 

First. 

Second. 

Third. 

Normal  Angle. 

counted. 

0 

, 

0 

/ 

0 

/ 

0         / 

m 

m' 

\ 

137 

S4 

53 

54 
332 

IS 

332 
249 

21 

mm'        8231*- 
m'a'       48  45! 

1  +  2  =  3 

2+2^4 

a' 

2 

6 

6£ 

36 

201 

5 

a'm"      48  44^ 

•m" 
m'" 

I 

2 

317 
334 

S6* 

234 
*52 

5i 

22 

"8 

20| 

m"m'i'  82  28f 
»»'"«      48  44i 

1  +  2  =  3 
0+2=2 

a 

O 

867 

1  1 

103 

3»i 

21 

6 

am          4845! 

0+1  =  1 

m 

I 

112 

24 

54 

534 

332 

21 

The  angles  indicate  that  m,  m' , 
;//',  /«"'  are  faces  of  one  form  and 
that  a,  a!  are  faces  of  another  form 
truncating  the  first  as  in  Fig.  194. 
The  most  probable  angles  will  be 
determined  by  averaging.  For  in- 
stance, the  probable  angle  m-m"r 
may  be  the  average  of: 


FIG.   1 94. 


Minutes. 


2  m  a 
2m'"  a 
2  m'  a' 


=97  3 
=97  2 
=97  3 


2  m'"  a'  =97 

180  —  mm'  =  97 

o  —  m"m'"  =  97 

m  m'"    =  97 

m'.m"    =9J 


Times  to  be  counted. 

i     or     i 
i 

2 

n 

3 
3 
3 
3 


=  97°  29F 

ERRORS  DUE  TO  IMPERFECT  CENTERING. 

Let  E  denote  the  error. 

Let  a  denote  the  angle  between  the  crystal  edge  and  the  goni- 
ometer axis. 

Let  f  denote  the  angle  obtained. 

*  The  errors  in  the  first  four  are  already  doubled,  hence  they  can  be  allotted  only 
half  the  sum  of  their  quality  marks. 


2 

57g~ 

4 

"5* 

3 

85 

6 

171 

6 

187 

6 

177 

6 

179 

34 

1012 

72  CHARACTERS  OF  CRYSTALS. 

a2 
Then  m\\E=*--y=  [cos  45°  —  cos  (2r  +  45°)]. 

This  value  is  a  maximum  when  cos  (2^  -f-  45°)  is—  I,  that  is, 
when  Y  =  67^2°  for  which  E=  -f^  a1  (approximately)  and  a  mini- 
mum when  cos  (2  ?  +  45°)  =  +  i  or  Y  =  1 57^  for  which  £  =  TL  a2. 

This  error  is  guarded  against  by  the  coincidence  of  signals  from 
different  faces  of  the  zone. 

In  addition. 

Let  /  denote  angle  of  incidence  of  the  light. 

Let  d  and  d'  denote  the  distances  of  the  faces  from  the  axis. 

Let  r  denote  the  distance  to  the  light  source. 

Then  will  E=2=*smi. 
r 

This  diminishes  as  r  increases  or  i  decreases  and  is  zero  if  r  is 
infinite  or  d=  d' ;  that  is,  the  second  error  is  eliminated  by  an  in- 
finite distance  collimator,  and  even  without  it  is  only  about  9"  if 
d  —  d'  =  j4  mm.  and  r  =  I  o  m. 

SPECIAL  CASES  IN  MEASUREMENT. 

Narrow  faces  tend  to  broaden  the  image  by  a  phenomenon 
analogous  to  diffraction.  The  diminishing  eyepiece  may  obviate 
this,  but  as  the  error  is  proportionatell  to  the  eccentricity  of  the 
face,  in  such  a  case  each  edge  must  be  separately  centered. 

Transparent  Crystals  give  colored  images  due  to  total  reflections 
within  the  crystal,  if  the  crystal  and  telescope  are  turned  steadily 
in  one  direction  these  images  at  a  point  move  backwards  (angle  of 
least  deviation) ;  when  such  an  image  coincides  with  the  true  one 
it  is  only  necessary  to  alter  the  angle  of  incidence. 

Finely  Striated  Faces  yield  a  bright  colorless  image  with  colored 
images  on  each  side.  The  bright  image  is  due  to  the  plane  tan- 
gent at  the  edges  of  the  striae  and  only  when  an  image  is  obtained 
after  the  plane  has  been  turned  into  some  other  zone  is  it  proved 
that  there  is  a  definite  crystal  face  underneath  the  striae. 

Bent  or  Cracked  Faces  yield  distorted  or  manifold  images.  An 
risblende  eyepiece  will  limit  the  reflection  to  a  selected  best  por- 
tion, or  an  approximate  measurement  may  be  obtained  from  the 
outer  member  of  each  group,  or  by  measuring  in  several  zones 

f  Story  Maskelyne's  Crystallography,  p.  414 
\  Ibid,  402. 

§  For  other  reasons  i  is  commonly  50°  to  60°. 
I!  Mull.  Soc.  Min.  de  France,  I.,  35. 


GEOMETRICAL   CHARACTERS.  73 

to  the  opposite  face  the  properly  oriented  portion  may  be'  found. 
The  best  way  is  to  use  another  crystal. 

7 win  Crystals  and  Composite  Crystals  in  approximately  parallel 
position  will  give  good  images  in  part  of  a  zone  and  in  the  rest  the 
images  will  be  a  little  out  of  center. 

Dull  Faces*  may  be  coated  with  thin  varnish  or  cover  glasses 
may  be  glued  to  them  or  the  extra  lens  may  be  used,  the  col- 
limator  slit  narrowed  to  the  smallest  width  which  will  give  an  illu- 
mination and  the  position  of  BRIGHTEST  ILLUMINATION  of  both  faces 
recorded  several  times.  Traube's  attachment  to  the  collimator  of 
the  Fuess  goniometer^  shown  in  Fig.  195  may  be  used.  It  is 


FIG.  195. 

practically  a  little  extra  dark  room  ;  the  light  passing  through  the 
collimator  and  the  conical  tube  d  to  the  crystal.  The  cylinder  e 
shuts  out  much  of  the  extraneous  light  giving  a  darker  field  in 
which  the  luminous  signal  is  by  contrast  brighter.  Still  more 
light  can  be  shut  out  by  laying  a  card  over  the  top  of  e  or  better  by 
the  cap  gt  which  may  be  turned  to  diminish  the  window /to  a 
mere  slit. 

Small and 'nearly parallel faces which  yield  a  combined  image  may 
be  distinguished  by  changing  the  angle  of  incidence. 

Crystals  which  alter  in  the  air  may  be  protected  during  measure- 
ment by  replacing  the  crystal  plate  by  a  hollow  cylinder  £  at 
Fig  196,  supporting  a  hemisphere  z  and  plate  /,  the  whole  cov- 
ered by  a  little  glass  bottle  with  bottom  pressed  in  and  ground  to  fit 
air-tight  on  the  oiled  hemisphere.  In  the  channeled  bottom  is  placed 
sulphuric  acid  or  chloride  of  lime  for  deliquescent  crystals,  and 

*Groth  describes  an  elaborate  application  goniometer  "  das  Fuhlhebel  goniometer  " 
for  measuring  dull  crystals  Physikalische  Krystal.,  III.   Ed.,  pp.  604-608. 
\Neus  Jahrb.  f.    Mineralogie,     1894,  Bd.  II. 
\Jahrb.  d.  geol.  Reichsanstalt,  1884,  329. 


74  CHARACTERS  OF  CRYSTALS. 


FIG.  196. 

water,  etc.,  for  efflorecent  crystals.  The  bottle  is  steadied  by  the 
spring  clamp  A,  and  the  light  enters  and  emerges  normal  to  the 
plane  windows  of  the  bottle. 

Imbedded  crystals  may  be  measured  to  within  one  degree  by  means 
of  impressions  in  sealing  wax. 

THEODOLITE  OR  TWO-CIRCLE  GONIOMETERS. 

In  the  reflection  goniometers  described  all  faces  of  the  zone  which 
is  placed  perpendicular  to  the  graduated  circle  reflect  the  signal  and 
if  the  circle  is  considered  to  be  the  plane  of  projection  their  poles 
are  in  the  circumference  at  angles  apart  equal  to  those  meas- 
uring the  rotation. 

If  a  second  direction  of  rotation  at  right  angles  to  the  other  be 
added,  any  face  in  any  zone  may  be  referred  to  this  first  zone  by  the 
two  motions  necessary  to  bring  it  into  the  field,  the  one  giving  the 
point  where  its  meridian  cuts  the  primitive  circle,  the  other  giving 
the  number  of  degrees  on  this  meridian  from  the  primitive  circle. 
Czapski,  Goldschmidt*  and  v.  Federovv  have  described  such  instru- 
ments; that  of  the  latter,  shown  in  Fig.  197,  consists  of  a  telescope 
By  which  is  also  the  collimator.  The  signals  are  on  the  revolving 
disc  by  and  the  light  entering  at  the  focal  plane  of  the  objective 
passes  through  a  small  total  reflecting  prism  and  emerges  as  parallel 
rays.  There  is  an  extra  lens  /  to  bring  the  crystal  into  focus  or,  by 

*Zeit.f.  Kryst.  XXI.,  210-232. 


GEOMETRICAL   CHARACTERS. 


75 


FIG.  197. 

means  of  the  spring  clamp  h  and  the  rack  and  pinion  H  and 
K,  one  of  the  weaker  objectives  of  a  microscope  may^be  focussed 
'on  the  crystal.  The  eye-piece  has  an  adjustable  irisblende. 

The  stand  consists  of  a  tripod  with  a  bored  conical  head  in  which 
rests  the  graduated  circle  C  turned  by  d  and  read  by  the  fixed  ver- 
nier N,  using  clamp  e  and  tangent  screw/.  The  stand  also  sup- 
ports D,  the  carrier  of  the  vertical  circle,  which  can  be  clamped  to 
the  stative  by  g,  or  to  the  horizontal  circle  by  e. 

The  vertical  circle  is  a  complete  Wollaston  goniometer,  but 
rotates  with  the  horizontal  circle  when  e  is  clamped.  By  trans- 
ferring the  centering  apparatus  to  the  horizontal  circle  the  instru- 
ment may  be  used  as  a  goniometer  with  vertical  axis. 


CHAPTER  VI. 


CRYSTAL   PROJECTION   OR   DRAWING. 

For  the  purposes  of  calculation  and  for  a  comprehensive 
view  of  the  relations  between  the  faces,  the  latter  are  more  con- 
veniently represented  either  by  points,  Fig.  198,  as  described  pp. 
20-24,  under  stereographic  projections,  or  by  lines  as  in  the  so- 
called  Linear  projection.  For  description  and  illustration  some 
projection  which  actually  figures  the  shape  of  the  crystal  is  usually 
preferred. 

LINEAR  PROJECTIONS. 

In  Quenstedt's  linear  projection*  each  face  of  the  crystal  is  as- 
sumed to  be  moved  parallel  to  itself  until  it  cuts  the  vertical  axis 
at  a  unit's  distance  above  the  the  plane  of  the  basal  axes.  The 
line  in  which  the  face  then  intersects  the  plane  of  the  basal  axes 
represents  the  plane. 

For  example,  the  crystal  of  topaz,  Fig.  201,  is  orthorhombic  in 
crystallization  with  a  :  b :  ^=.529  :  I  :  .477.  The  basal  axes  are  drawn 
at  right  angles,  and  the  proportionate  lengths  .529  :  I  laid  off  upon 
them.  As  the  crystal  is  symmetrical  to  the  axial  planes  it  is  only 
necessary  to  determine  the  projections  of  the  faces  of  one  octant, 
say  the  upper  right  hand.  The  indices  of  these,  therefore,  are 
written,  then  their  reciprocals,  which  are  the  axial  intercepts  (p. 
12),  and  finally,  these  divided  by  the  third  term,  that  is,  the  inter- 
cepts of  the  faces  when  they  cut  the  vertical  axis  at  unity — 

Indices.  Reciprocals.  Unit  c  intercepts. 

in  ill  in 

110  III  001 

120  i^o  ooi^: 

223  I**  .  If i 

041  i\i  i\l 

*  Beitrdgen  zur  rechnenden  Krystallographie,  Tubingen,  1848. 

f  In  Goldschmidt's  Euthygraphic  Projection  the  faces  cut  a  unit's  distance  below 
the  basal  axes.  Ueber  Projection  u.  graph,  Krystal.,  p.  25. 

\  Prisms  are  all  projected  as  lines  through  the  center  parallel  to  the  projections  of 
pyramids  of  the  same  zone.  In  this  case  such  a  pyramid  as  121. 


GEOMETRICAL   CHARACTERS.  77 

The  points  on  a  and  b  corresponding  to  the  first  and  second 
term  of  each  unit  c  intercept,  are  then  connected  by  a  straight  line 
which  is  the  linear  projection  of  the  face.  Fig.  200. 

The  projection  of  the  edge  between  any  two  faces" is  evidently 
the  line  from  the  center  of  the  projection  to  the  intersection  of  the 
two  lines  representing  the  faces,  and  the  direction  of  that  edge  is 
the  line  from  the  intersection  to  the  unit  point  on  the  vertical  axis. 
Hence  all  faces  projected  in  lines  with  a  common  intersection  in- 
tersect in  parallel  edges  or  lie  in  one  zone.  Evidently  also  a  face 
lying  in  two  zones  will  be  projected  in  the  line  through  the  two 
common  intersections  or  zonal  points. 

The  zonal  equations  may  be  here  used  as  described,  pp.  17-20, 
and  calculations  may  be  made,  though  less  conveniently  than  with 
the  stereographic  projection. 

PROJECTIONS  IN  PARALLEL  PERSPECTIVE. 

If  the  eye  be  conceived  to  be  at  an  infinite  distance  the  visual 
rays  become  parallel,  and  all  parallel  lines  remain  so  *  when  pro- 
jected,-and  the  proportionate  lengths  into  which  any  line  is  divided 
are  .not  changed,  although  the  line  may  be  foreshortened. 

ORTHOGRAPHIC  PARALLEL  PERSPECTIVE. 

The  plane  of  projection  is  at  right  angles  to  the  visual  rays.  If 
the  plane  is  the  base  the  basal  axes  are  full  length,  but  if  it  is  at 
right  angles  to  the  vertical  axis  the  axis  a  in  monoclinic  crystals 
will  be  foreshortened  to  a  sin  p  and  in  triclinic  crystals  a  will  be- 
come a  sin  ft  and  b  will  become  b  sin  a. 

The  faces  of  the  zone  normal  to  the  plane  of  projection  will  be 
projected  as  lines  bounding  the  figure  at  their  true  angles.  The 
edges  between  other  faces  will  be  parallel  both  to  the  edge  projec- 
tions (see  above)  of  the  linear  projection  on  the  same  plane  or  to 
the  tangent  to  the  primitive  circle  at  the  intersection  with  the  zone 
circle  of  the  two  faces.  For  instance,  in  Fig.  199,  the  direction  of 
the  orthographic  projection  of  any  edge,  say  [041,  223],  is  both 
that  of  the  line  Oc,  Fig.  201,  and  of  the  tangent  at  O,  Fig.  198. 

*  Lines  parallel  in  projection  are  not  necessarily  so  in  the  crystal,  for  all  lines  in  a 
plane  through  two  rays  are  projected  in  the  same  line. 


CHARACTERS  OF  CRYSTALS 


FIG.  200. 


FIG.  201. 


FIG.  202. 
PROJECTIONS  OF  A  TOPAZ  CRYSTAL. 

198.  Stereographic.  199-  Orthographic  Parallel. 

200.  Linear.  201.  Clinographic  Parallel. 

202.  Linear  upon  axial  cross. 


GEOMETRICAL  CHARACTERS. 


79 


CLINOGRAPHIC  PARALLEL  PERSPECTIVE. 

The  plane  of  projection  is  oblique  to  the  visual  rays.  This  method 
gives  an  appearance  of  solidity  and  usually  is  made  upon  a  vertical 
plane,  the  point  of  sight  being  to  the  right  and  above  the  crystal. 

The  Axial  Cross,  or  perspective  view  of  the  axes,  is  first  prepared. 

If  the  angle  to  the  right',  or  horizontal  angle,  is  denoted  by  d  and 
the  angle  of  elevation  by  e,  then  the  following  relations*  exist  be- 
tween these  angles  and  the  projected  lengths  and  angles  of  the 
isometric  axes  in  which  the  notation  is  as  in  Fig.  203. 


f^V.i^ —  cos2^  cos2  e  ;  OB  =  ^/  i  —  sin2  d  cos2  e  ;  OC=  cos  e 
cot  AOC=  —  d  sin  e ;  cot  BOC=  —  tan  <5  sin  e ;  cot  8  =     I 


cot  AOC 


cot  BOC 


COS  SiUL  = 


OB*  -  OC*)  (OB1  +  OC*  -  OAZ) 
~  ' 


cos  BOC  =  — 


\/(OA2  +  OC*  —  OB*)  (OA*  4-  OB?  —  OC*) . 
2  OB.OC 


The  following  table  gives  a  series  of  projections,  of  which  num 
bers  3,  4,  5,  6  and  10  are  usually  preferred. 


Number." 

Angles  of 
Revolution. 

Angles 
between  projected 
Axes. 

Foreshortened 
lengths  of  axes  for 
true  length  =  i. 

Approximate 
Proportionate 
lengths. 

6 

e 

^<?C 

BOC 

OA 

OB 

OC 

OA 

OB 

OC 

i 

n°  35' 

Oo49, 

93°  58' 

90°  10' 

.200 

.980 

•  999 

10 

49 

5° 

2 

3 

14°  35' 
180  26' 

lO  I7/ 
60  20' 

94°  55' 
1080  19' 

900  20' 
920  06' 

.250 
•239 

.968 
.828 

•  999 
•994 

8 
29 

IOO 

32 

120 

4 

1  80  26' 

70  1  1/ 

i  ioO  34/ 

920  23' 

•338 

•950 

.992 

36 

IOO 

i°5 

c 

1  80  26' 

90  28' 

ii60  i7/ 

93°    8' 

•353 

•95° 

.986 

37 

IOO 

104 

6     190  16' 

20  I4/ 

960  23' 

900  47/ 

•333 

•944 

.998 

6 

17 

18 

8 

27°  59' 

9°50/ 
35°  1  6' 

1070  49' 

1200 

95°  ii' 

1200 

•493 
.817 

.887 
.817 

•  985 
.817 

5 

i 

9 

i 

10 

i 

9 

200  42' 

190  28' 

1310  25' 

970  1  1/  1  .471 

•943 

•943 

i 

2 

2 

10 

13°  38' 

13°  15' 

133°  24' 

930  1  1/  1  .324 

•973 

i 

3 

3 

ii 

ioO  10' 

IOO 

1340    6' 

9i°  47' 

.246 

•985 

.985 

i 

4 

4 

*  Story-Maskelyne's  Crystallography,  p.  480.     Jos.  Barrett,  Lchigh  Quarterly. 


\ 

\ 
\ 


8o  CHARACTERS  OF  CRYSTALS. 

In  drawing  the  cross,  OC  is  vertical  and  the  directions  of  OA 
and  OB  result  from  the  angles  AOC  and  BOC.  The  proportionate 
lengths  may  be  laid  off  at  any  scale  desired. 

When  cot  d  and  cot  s  are  simple  numbers  as  in  Nos.  3  and  5  in 
which  cot  £=3  and  cot  e  =  g  and  6  respectively,  the  projection 
of  the  axial  cross  may  be  obtained  as  follows : 

A  horizontal  line  ss,  Fig.  203,  is  bisected  by  a  perpendicular,  and 
perpendiculars  are  drawn  as  indicated  at  the  ends  and  at  points 
/  and  tf  and  distances  laid  off  so  that 

O£=  Of  =  sg=*  — 5—  and  s'e  =  Os  — - 

COt  d  COt  £ 

(in  the  figure  cot  d  =  3,  cot  e  =  6). 

The  point  *  determines  the  line  eO 
and  AA'\  the  projection  of  the  front 
horizontal  axis  is  the  portion  of  this 
line  which  is  included  between  the 
perpendiculars  at  t  and  f.   The  point    «• 
g  determines  the  radius  Og,  which  is    ^., 
the  length  of  half  the  vertical  axis  CO.      II 
For  the  projection  of  the  third  axis 
draw  Af  parallel  to  ss',  draw/<9,  and 
from  the  intersection  v  draw  vB  par- 
allel to  ss'.     BB'  completes  the  de-  FlG  203 
sired  projection  or  axial  cross. 

The  axial  cross  for  crystals  of  other  systems  is  derived  from  the 
isometric  cross  by  assuming  the  latter  to  be  the  projection  of  lines 
each  equal  in  length  to  that  axis  of  the  crystal  which  extends  con- 
ventionally from  left  to  right  (ft). 

In  the  TETRAGONAL  and  ORTHORHOMBIC  crosses  the  proportion- 
ate values  of  the  axes  c  and  a  are  laid  off  on  CO  and  AA'  re- 
spectively. In  Zircon,  for  instance,  the  vertical  axis  is  made  ^T  of 
CO. 

In  the  MONOCLINIC  crosses,  Fig.  204,  the  vertical  axis  is  a  propor- 
tionate length  on  CC,  but  A  A  is  the  projection  of  a  line  at  right 
angles  to  CO  and  the  direction  of  the  clino  axis  is  first  found  by 
laying  off  Or=OC  cos  /?  and  On—OA  sin  and  completing  the  par- 
allelogram OAtr.  The  proportionate  value  of  a  is  then  laid  off 
upon  the  diagonal  Ot. 


1 8 


GEOMETRICAL  CHARACTERS. 


81 


This  and  other  projections  are  quickly  obtained  by  using  a  metal 
quadrant.*  Fig.  204  the  center  of  which  is  at  B.  The  edge  AB 
and  the  arc  AC  are  tapered  to  a  thin  edge  for  greater  exactness  in 
marking.  With  AB  ten  centimeters  long  the  results  will  be  cor- 
rect within  the  limits  of  a  drawing. 


FIG.  204. 

A  scale  line  OS  is  drawn  at  will  from  the  center  0  and  axial 
lengths  are  transferred  directly  from  AB  to  the  scale  line  approxi- 
mately to  the  third  decimal.  Sines  and  cosines  are  transferred  as 
follows.  If  the  edge  BC  and  the  scale  line  are  made  coincident 
and  then,  by  use  of  a  triangle,  the  quadrant  is  slid  along  in  a  direc- 
tion at  right  angles  thereto  (BC  remaining  parallel  to  the  scale  line) 
until  the  scale  line  cuts  the  arc  at  the  given  degree  and  the  approxi- 
mate minute,  the  intercept  on  the  scale  line  will  be  the  sine.  Simi- 
larly with  the  edge  AB  coincident  with  the  scale  line  and  a  motion 
at  right  angles  thereto,  the  intercept  on  the  scale  line  will  be  the 
cosine. 

All  measurements  are  transferred  to  other  lines,  by  lines  parallel 
to  the  direction  between  the  ends  of  .the  scale  line  and  the  unit 
line.  For  instance,  in  a  monoclinic  crystal  in  which  a  :  b  :  c= 
1.092  :  I  :  0.589  and  ,3=74°  10'.  Ok  is  sine  74°  10'  when  radius  is 
OS  and  transferred  to  OA  is  On\  Odis  cosine  74°  10'  when  radius 
is  OS  and  transferred  to  OC  is  Or\  by  completing  the  parallelo- 
gram t  and  Ot  result.  For  the  axial  lengths,  Oi  is  1.092  times  OS 
and  Oe  is  0.5  £9  times  OS  and  transferred  are  respectively  Oa= 


*  Described  A.  J.  Moses,  Am.  Journ.  Science  ',  I.,  June,  1896. 


82 


CHARACTERS  OF  CRYSTALS. 


In  the  triclinic  crosses,  Fig.  205,  the  same  methodjis  carried  fur- 
ther, for  instance:  The  constants  for  axinite  are  a  :  b :  c—  0.492" 
I  :o.479,  a  f\  <r=/2=9i°  52',  b  A  ^=^=82  54'  (100)  /,  (oio)= 
131°  39'. 

Vertical  Axis.    Make  Cfc=  (9£7x  479- 

Macro  Axis.  Make  Oe=OB  sin  131° 
39',  and  <9^=  <9^4  cos  1 3 1  °  39';  com- 
plete the  parallelogram  <^(9^^.  Make 
Ot  =  0n  sin  82°  54'  and  (9^r=(7Ccos 
82°  54'  ;  complete  the  parallelogram 
rOxb.  Then  is  0£  the  projection  of 
one-half  the  desired  axis. 

Brachy  Axis.  Make  01=  OC  cos 
91°  52',  and  Op=OA  sin  91°  52'. 
Complete  the  parallelogram  ;  pOIt 
make6>#=  0.492  =  <9/;  then  is  Oa 
the  projection  of  one-half  the  desired  axis.  These  results  are  more 
quickly  made  with  the  quadrant  previously  described. 

In  the  HEXAGONAL  CROSSES,  Fig.  206,  the  proportionate  value  of 
c  is  laid  off  on  CO  and  the  basal  axes  are  derived  as  follows  : 

Make  Op=  6^4^.1.732  ;  draw  pB 
and  pB' ;  bisect  Op  by  a  line  parallel 
to  BB'\  then  are  OB,  Oa  and  Oa^ 
the  projections  of  desired  semi-axes. 


DETERMINATION 
OF  EDGES. 


OF  THE   DIRECTION 


FIG.  206. 


The  unit  form  always  results  from 
joining  the  extremities  of  the  axial 
cross  by  straight  lines  and  other  sim- 
ple forms  are  easily  drawn  by  meth- 
ods which  suggest  themselves, for  in- 
stance, the  unit  prism  bylines  through 
the  terminations  of  the  basal  axes 
It  is  always  possible,  also,  to  obtain 
two  points  of  any  edge  by  actually  constructing  the  two  planes 
and  rinding  the  intersection  of  their  traces  in  two  axial  planes. 
The  method,  however,  is  cumbersome. 

In  all  systems  the  projected  intersection  may  be  most  easily 
found  by  the  following  method  :     A  linear  projection  of  the  faces 


parallel  to  the  vertical  axis. 


GEOMETRICAL   CHARACTERS.  83 

is  made  (Fig.  202)  upon  the  basal  axes  of  the  axial  cross  precisely 
as  described  for  the  ordinary  linear  projection.  One  point  of  the 
edge  between  any  two  planes  is  the  unit  point  on  the  vertical  axis ; 
another  is  the  intersection  of  the  linear  projections  of  the  two 
planes  and  the  line  connecting  these  is  the  edge.  For  instance, 
Oc,  Fig.  202,  is  the  direction  of  the  edge  [041,  223]  and  is  so  drawn 
in  Fig.  201. 

CONSTRUCTION  OF  THE  FIGURE. 

The  edges  thus  formed  must  be  united  in  ideal  symmetry,  yet  so 
as  to  show,  as  far  as  possible,  the  relative  development  of  the  forms. 

A  second  axial  cross  is  drawn  parallel  to  that  used  in  determin- 
ing the  edge  directions  and  these  are  transferred  by  triangles,  care 
being  taken  that  all  corresponding  dimensions  are  in  their  proper 
proportions  and  in  accord  with  the  planes  of  symmetry.  Gen- 
erally it  will  be  best  to  pencil  in  and  verify  the  principal  forms 
and  later  work  in  the  minor  modifying  planes. 

The  back  (or  dotted)  half  of  many  crystals  can  be  obtained  by 
marking  the  angles  of  the  front  half  on  tracing  paper,  turning  the 
paper  in  its  own  plane  1 80°  and  pricking  through.  This  is  also  a 
test  of  accuracy,  for  the  outer  edge  angle  for  angle  should  coin- 
cide. 

TWIN  CRYSTALS. 

These  have  two  set  of  axes,  the  second  so  related  to  the  first  that 
it  corresponds  to  a  revolution  of  1 80°  about  the  twin  axis  or  line 
normal  to  the  twining  plane. 

The  two  individuals  may  be  in  apposition,  that  is,  the  twin  plane 
coinciding  with  the  combination  face.  In  this  case  the  twin  axis 
which  passes  through  both  centers  normal  to  the  twin  plane  will 
be  bisected  by  the  latter. 

In  interpenetrating  twins  the  two  centers  may  coincide  or  be 
near  together.  The  orientation,  however,  will  be  as  in  the  former 
case. 

Given  the  axial  cross  of  one  crystal  to  find  that  of  a  second  crystal 
in  apposition  thereto.* 

Let  OA  OB  OC,  Fig.  207,  be  the  axial  cross  of  the  first  crystal 
and  HKL  the  twinning  plane.  To  find  first  the  point  Zat  which 
the  normal  from  0  cuts  HKL,  draw  H'L  and  HLf  parallel  AC 
and  draw  KL"  and  K' L  parallel  BC.  Complete  the  parallelograms 

*Groth,  Physikalisoche  Krystallographie,  p  599,  III  Ed. 


84 


CHARACTERS  OF  CRYSTALS. 


FIG.  207. 

OH' ML'  and  OK'NL"  and  draw  their  diagonals  OM  and  ON  and 
from  the  intersections  of  these  with  HL  and  LK  draw  RK  and  SH 
respectively ;  their  intersection  is  the  desired  point  Z. 

Because  the  crystals  are  in  apposition  prolong  OZ  till  ZO'  =  OZ, 
then  is  0  the  center  of  axes  for  the  second  crystal  and  as  the  face 
HKL  is  common  0'Ht  0' K  and  O'L  are  in  direction  and  length 
the  coordinates  of  this  face  on  the  new  axes. 

The  unit  lengths  will  be  found  by  drawing  AA't  BE'  and  CO 
parallel  to  O'Z. 

All  constructions  on  the  axes  of  the  second  crystal  follow  the 
rules  previously  given. 


OPTICAL   CHARACTERS.  87 

of  time.  The  ray  front  in  the  first  medium  at  the  expiration  of 
this  time  would  be  TE,  E  being  a  point  of  the  front  just  impinging 
on  AB,  therefore  also  a  point  of  the  new  ray  front,  that  is  of  the 
tangent  plane  ES.  The  refracted  ray  is  therefore  *  OS  to  the 
point  of  tangency.  OYis  the  corresponding  reflected  ray. 
In  the  triangles  OETand  OES,  Fig.  208, 


>,  os 

OE  =  -  —  ^r~  =  -r^  and  OE  = 


•  x-\    j  *  rt-\     -  •  •        C*««Vt      \S  •*—*    —  '  •  x-~\    7""  {~*    "'      '"'         * 

sin  OET      sin  2  sm  OES       sin  /> 

hence, 

7',       sin  i 
v^      sin  p 

Usually  the  ratio  recorded  is  that  of  the  crystal  with  respect  to 
air.  Denoting  the  velocity  in  air  by  v  and  the  indices  of  the  outer 
medium  and  crystal  with  respect  to  air  by  nL  and  n,  we  have 

v  -  v 

n.  =  —  and  n  =  —     . 
v\  vz 

and  substituting  these  values  we  have 

sin  i 


n  = 


1  sin  p 
There  is  no  refraction  for  normal  incidence,  for  sin  z  —  o,  hence 

o 

sin  p 
which  is  only  possible  if  p  —  o. 

With  a  plane-parallel  plate  the  ray  emerging  is  parallel  to  the 
entering  ray,  for  at  entrance,  Fig.  210, 

sin  i  sin  z, 

and  at  emergence  n^  =  n 


sin  p  sin^ 


#j      sin  p       sm  ^ 

but  /t  =  p     hence  i=  p^ 

An   approximate   determination    of   the 
index  of  refraction  may  be  made  by  meas- 
FIG.  210.  uring  the  displacement  of  the  focal  distance 

*  This  is  called  the  Huyghens  construction,  each  point  of  the  border  surface  becom- 
ing a  new  center  of  propagation  of  light.  A  still  simpler  construction  is  that  of  Snel- 
lius,  Fig.  209,  ^  is  a  sphere  with  radius  the  index  of  refraction  of  outer  medium,  R.^  a 
sphere  with  index  of  crystal.  Prolong  IO  to  T.  Draw  TS  parallel  the  normal  ON, 
then  is  OS  the  refracted  ray,  and  OK  the  corresponding  reflected  ray. 


88  CHARACTERS  OF  CRYSTALS. 

of  a  microscope  caused  by  the  interposition  of  a  known  thickness  of 
crystal  as  described,  p.  120,  but  usually  one  of  the  following  meth- 
ods will  be  employed. 

Determination  of  Index  of  Refraction  by  Prism  Method. 

Let  AOC,  Fig.  211,  be  the  section  of  the  prism  at  right  angles 
to  the  refracting  edge  O.  About  O  describe  the  circles  R^  and  R2  with 
radii  proportionate  to  the  indices  of  refraction  of  the  outer  medium 
and  prism  respectively.  Let  10  be  the  incident  ray,  then,  by  the 
construction  of  Snellius  (foot  note  p.  89),  is  OS  the  direction  of  the 
ray  in  the  prism.  From  Sdraw  SP  normal  to  the  surface  CO,  then 
is  OP  the  direction  of  the  ray  on  emergence.  Denoting  the  prism 
angle  by  /  —  NON1  the  total  deviation  by  d  =  TOP,  the  incident 
angle  by  z  =  TON,  and  the  angle  of  refraction  at  the  second  surface 
by  Pl=PONl. 

<5  +  7  =  TOP  +  NOW  =  2  TOP  +  PON  +  TON1, 
i  +  ^  =  TON+  PON1  =  2  TOP  +  PON  +  TON1 
hence, 

3  +  X  =  *  +  Pl  or  d  =  i  +  Pl  -  x. 

d  -4-  y 

This  value  of  d  is  least  *  when  i=  />lt  then  <5  =  2  z  —  7  z  =  -  —  . 

When  z  =  />!  the  ray  RS,  Fig.  212,  within  the  prism  must  be  nor- 
mal to  BD,  the  bisectrix  of  the  refracting  angle,  therefore  will  NRS 
=  ABD  or  p  =  y2  /.  Hence  substituting  in  the  formula  for  index 

r       f        ,.-  Sm  *  u 

of  refraction  ;/  =  n,~  —  ,  we  have, 
sm/> 


PRACTICAL  MANIPULATION.  —  Two  perfect  faces  of  a  clear  trans- 
parent crystal  are  required,  making  such  an  angle  (40°  to  70°)  with 
each  other  that  at  the  second  surface  the  ray  is  incident  at  less  than 
the  angle  of  total  reflection,  p.  90,  or  with  a  larger  angle,  the  prism 
may  be  immersed  in  a  strongly  refracting  liquid  in  a  parallel  walled 

*The  arc  TP  or  rf,  Fig.  211,  cut  by  SP  and  ST,  is  least  when  they  make  equal 
angles  with  OS  that  is  when  the  first  deviation  i  —  p  equals  the  second  pl  —  il  for 
with  any  other  position  of  IO  the  point  7'  moves  a  certain  number  of  degrees  and  one 
of  the  arms  TS  or  PS  approaches  OS,  becoming  less  oblique  and  cutting  off  a  part  of 
TP,  the  other  recedes  becoming  more  oblique  and  adding  a  larger  arc  to  TR,  hence 
the  combined  change  yielding  a  larger  value  for  J. 


OPTICAL   CHARACTERS. 


89 


glass  vessel.  If  necessary,  faces  may  be  ground  at  the  proper  angles. 
The  other  faces  of  the  crystal  should  be  coated  with  lamp  black. 


FIG.  211.  FIG.  212. 

The  most  satisfactory  instrument  is  a  goniometer  with  vertical 
axis  seep.  66-74.  The  angle  /  is  centred  and  measured  as  de- 
scribed, p.  69.  The  telescope  is  then  clamped  at  7]  Fig.  2  12,  directly 
opposite  the  collimator  K  and  a  reading  made,  the  crystal  is  moved 
by  the  centring  screws  so  that  the  edge  B  is  a  little  beyond  the 
centre,  and,  the  telescope  and  graduated  circle  remaining  clamped, 
is  revolved  into,  for  example,  the  position  shown  in  the  figure. 
The  telescope  is  then  undamped  and  turned  towards  the  left  until 
the  image  is  in  the  field.  The  crystal  is  turned  towards  the  right 
and  if  the  image  moves  it  is  recentred  by  a  movement  of  the  tele- 
scope towards  the  right,  and  this  double  motion  is  continued  until 
the  image  appears  for  a  moment  to  be  stationary  and  then  starts 
to  move  in  the  opposite  direction. 

The  position  of  rest  Tr  is  the  position  of  least  deviation  and 


Monochromatic  light*  is  essential. 

*  Certain  solids  vaporized  in  the  flame  of  a  Bunsen  burner  emit  light  essentially 
monochromatic.  Three  very  commonly  used  are  RED  /,  =  0.000670  Lithium  sul- 
phate, YELLOW  ?„  =  0.000589  Sodium  sulphate,  Green  ?„=  0.000535  Thallium  sul- 
phate. Purer  light  may  be  obtained  by  using  portion  of  a  spectrum.  A.  E.  Tutton  de- 
scribes an  instrument  for  thus  producing  light  of  any  desired  wave  length  of  greater 
brilliancy  than  that  yielded  by  a  colored  flame.  Proc.  Royal  Soc.  1894,  v.  55,  p.  ill. 

The  production  of  monochromatic  light  by  absorption  of  the  other  colors  is  not  pos- 
sible. Blue  cobalt  glass  permits  to  pass  only  blue  and  extreme  red.  Certain  solutions 
absorb  certain  rays;  aniline  blue  absorbs  yellow;  permanganate  of  potash,  green  ;  sul- 
phate of  copper,  red  ;  chromate  of  potash,  blue.  The  light  passed  through  a  series  of 
these  may  become  essentially  monochromatic,  e,  g.t  Sulphate  of  copper,  aniline  blue 
and  chromate  of  potash,  leave  green. 


CHARACTERS  OF  CRYSTALS. 


If  the  instrument  used  does  not  permit  of  independent  rotation 
of  the  crystal  the  position  of  minimum  deviation  and  the  reading 
T'  are  obtained  by  alternate  movements  of  the  telescope  and  di- 
vided circle  and  the  reading  of  the  collimator  or  T  is  obtained  last. 

Determination  of  Index  of  Refraction  by  Total  Reflection. 

When  the  index  of  refraction  n±  of  the  outer  medium  is  greater 
than  n  of  the  crystal  there  is  a  so-called  "critical"  angle  of  inci- 
dence for  which  the  angle  of  refraction  is  90°  ;  that  is,  the  refracted 
ray  travels  along  the  border  surface. 

If  p  =3  90°,  sin  p  =  I  ;  hence,  n  —  n^  sin  i  or  sin  2  =  — . 

n\ 

For  any  angle  of  incidence 
greater  than  this  the  light  is  to- 
tally reflected. 

The  following  is  the  construc- 
tion of  Snellius:  If  AB,  Fig.  21 3, 
is  the  border  surface,  R±  and  Rz 
circles  with  radii  proportionate  to 
the  indices  of  refraction  of  the 
first  and  second  substances,  then 
for  the  critical  angle  the  direction 
of  the  refracted  ray  must  be  OP 
and  from  the  tangent  at  P  results  the  direction  lOTot  the  limit 
incident  ray. 

According  to  the  relative  position  of  the  observation  telescope 
and  the  incident  light  two  different  results  are  obtained. 

i°.  TOTAL  REFLECTION  PROPER.     If  diffused  light  is  admitted  in 
the  quadrant  AN,  Fig.  213,  and  the  telescope  axis  is  in  the  direction 
OW,  all  rays  incident  at  less  than  the  critical  angle  are  in  part  re- 
flected, e.  g.t  0  Y  along  OM  and  in  part  penetrate  the  crystal ;  while 
all  rays  incident  at  more  than  the  critical  angle  are 
entirely  reflected ;  that  is,  the  telescope  field  re- 
ceives on  one  half  totally  reflected  rays,  on  the 
other  partially  reflected  rays ;  between  these  is  a 
sharp  line,  Fig.  214,  which  is  the  intersection  of 
the  focal  plane  of  the  telescope  with  a  limit  sur- 
face or  cone  the  vertex  of  which  is  at  O  and  the 
elements  of  which  make  the  critical  angle  with 
the  normal  to  the  reflecting  surface.     This  limit  line  is  a  curve,  but 


FIG.  214. 


OPTICAL   CHARACTERS. 


FIG.  215. 


within  the  limits  of  the  field  of  the  telescope  is  essentially  straight. 
2.  GRAZING  INCIDENCE,  or  Observation  of  Transmitted  Light.*  If 
the  light  is  shifted  to  the  quadrant  AL,  or,  which  is  equivalent,  the 
light  remains  in  y^Vand  the  telescope  axis  is  made  to  coincide 
with  OT and  the  transmitted  rays  are  viewed,  then,  assuming  the 
faces  at  entrance  and  emergence  to  be  parallel, 
the  incident  and  emerging  rays  are  parallel. 
All  rays  incident  at  less  than  the  critical  angle 
are  partially  transmitted,  but  all  of  greater  angle 
are  totally  reflected  at  the  first  surface ;  the  tele- 
scope field  is  on  that  side  dark,  Fig.  215,  but 
on  the  other  side  is  illuminated  by  the  rays  in- 
cident at  less  than  the  critical  angle. 

In  the  Kohlrausch  apparatus  the  substance  is  supported  in  a 
liquid  of  higher  index  of  refraction  than  the  crystal,  in  such  a  way 
that  the  reflecting  surface  is  vertical  and  the  crystals  can  be 
rotated  about  a  vertical  line  in  the  reflecting  surface.  The  crystal 
holder  may  be  simply  a  metal  plate  with  a  window-like  opening 
bisected  by  a  platinum  wire  and  adjusted  once  for  all  so  that  the 
back  surface  is  in  the  desired  position  of  the  reflecting  face  and 
the  wire  coincides  with  the  vertical  cross-hair  of  the  telescope.  It 
is  only  necessary  with  this  to  fasten  the  crystal  face  over  the  win- 
dow. A  more  elaborate 
holder  permits  rotation  of 
the  crystal  in  its  own  plane 
and  other  adjustments. 

The  rotation  is  observed 
by  a  horizontal  telescope 
placed  in  the  direction  719, 
Fig.  216,  normal  to  a  plane 
front  of  the  vessel  holding 
the  liquid,  and  the  rotation 
is  recorded  upon  a  gradu- 
ated circle.  The  best  posi- 
tion for  the  light  is  found 
by  trial.  When  the  sharp 
limit  linebetween  thetotally 

*  Apparatus  for  observation  of  the  transmitted  ray  with  liquids  were  constructed  by 
Christiansen,  Pogg.  Ann.,  1871,  143,  p.  250  and  others;  and  for  solids  by  Quinke 
Zeit.  f.  Jfryst.,  1879,  4,540. 


FIG.  216. 


92  CHARACTERS  OF  CRYSTALS. 

and  partially  reflected  rays  has  been  made  to  coincide  with  the 
vertical  hair  of  the  telescope  the  light  and  screen  are  moved  to  the 
opposite  side  and  the  plate  is  rotated  until  the  limit  line  is  again 
obtained  and  centered. 

Since  the  angle  between  the  telescope  axis  and  the  normal  is  the 
critical  angle  the  rotation  NON1  =  22  whence  by  ;/  =  ;/A  sin  i  the 
index  of  the  solid  results. 

A  simpler  apparatus  is  made  as  an  attachment  to  the  No.  2  Fuess 
Goniometer*  and  is  shown  in  Figs.  217,  218.  The  pin  a  fits  in 
place  of  the  pin  of  the  usual  crystal  plate  of  the  instrument ;  pro- 
vision for  approximate  adjustment  is  made  but  the  accurate  adjust- 
ments of  the  Fuess  instrument  are  the  principal  reliance. 

To   secure   approximately  constant  temperature  the  holder   is 

covered  by  a  box  of  asbestos  with 
proper  openings.  The  mineral  is  at- 
tached to  the  little  plate/,  and  the  con- 
trol mineral  at  g.  The  rough  adjust- 
ments are  made  by  the  eye,  so  that 
the  necessary  rotation  can  be  secured  ; 
then  the  finer  adjustment  is  made  with 
the  Fuess  centring  screws,  and  the 
vessel  filled  with  the  refracting  liquid. 
FIG.  217.  FIG.  218.  The  cover  box  is  then  put  on,  the  light 

adjusted,    and    the    boundary    found 

and  centred.  After  standing,  say  V2  hour,  with  light  burning,  the 
boundary  is  recentred  and  this  repeated  till  no  change  takes 
place.  The  usual  readings  are  then  made,  after  which,  without 
disturbing  vessel  or  cover,  the  control  mineral  f  is  raised  into  the 
field  by  the  vertical  screw  and  readings  obtained  from  it  From 
these  the  index  results  by  the  formula 

,  sin  t 
n  =  n'—. 

sin  P 

in  which 

i  =  half  the  angle  of  rotation  of  the  mineral  tested. 
F=    "      "       "       "         "          "     "     control  mineral. 
n'  =  index  of  refraction  of  the  control  mineral. 

*A.  J.  Moses  and  E.  Weinschenk,  Zeit  f.  Krystallog.,  XXVI.,  150  and  S  cf  M. 
Quarterly,  XV ill.,  p  12. 

f  For  a  control  mineral  fluorite  is  very  suitable,  as  its  refraction  has  been  carefully 
determined,  and  further,  it  is  isotropic,  has  a  low  index  of  refraction  and  is  easily 
obtained. 


OPTICAL    CHARACTERS. 


93 


These  instruments  require  either  monochromatic  light,  or  sun- 
light may  be  used  if  the  eye-piece  is  replaced  by  a  spectroscope, 
in  which  case  at  the  proper  angle  the  light  of  all  colors  will  be 
totally  reflected  and  the  field  will  show  on  one  side  a  bright 
spectrum,  on  the  other  a  relatively  dark  spec- 
trum, separated  by  a  line  oblique  to  the  verti- 
cal hair,  Fig.  219.  The  critical  angles  cor- 
responding to  the  Frauenhofer  lines  can  be 
successively  determined  by  bringing  the  points 
of  intersection  of  these  lines  with  the  limit 
line  into  contact  with  the  vertical  hair. 

The  Liebisch  apparatus,  Fig.  220,  employs 
a  glass  prism  P  of  high  index  of  refraction, 
firmly  mounted*  with  the  refracting  edge 
vertical  and  one  face  normal  to  the  axis  of  the  holder.  Diffused 
monochromatic  light  is  admitted  at  the  side  AB,  Fig.  221,  reflected 
at  the  second  side  JSCand  emerges  at  the  third  side  AC. 

The  edges  of  the  prism  are  first  made  vertical;  the  crystal  is 
then  glued  to  the  cap  z  and  adjusted  by  the  screws  q  so  that  a 
collimator  signal  from  the  crystal  remains  fixed  in  the  telescope 
during  a  complete  rotation  by  T.  By  the  centring  screws  y  y  of  the 
goniometer  a  central  line  of  the  crystal  is  made  to  coincide  with 
the  goniometer  axis. 


FIG.  219. 


FIG.  220. 


*As  made  by  Fuess  two  prisms  with  indices  1.6497,  '-7849  are  furnished. 


94  CHARACTERS  OF  CRYSTALS. 

The  prism  is  then  moved  into  contact  with  the  crystal,  close 
contact   being   secured   by  a  drop  of  strongly  refracting   liquid. 
The  determination  of  the  index  of  the  plate  requires  the  meas- 
urement of  «j  the  index  of  the  prism,  of «  =  ACB,  the  refracting 
v  angle  of  the  prism,  and  of  d,  the  de- 

\  viation  of  the  emerging  ray  from  the 

\  normal  to  the  face  of  emergence. 

To  determine  o  three  readings  are 
'•  ^  needed  :  First t  the  telescope  and  col- 

limator    are    placed    opposite   each 
other,  say  at  T  and  V  respectively. 
Second,   the   telescope    is   arbitrarily 
\  moved  to  some  position  L  and  the 

carrier  turned  until  the  limit  line  is 
centred.     Third,  the  carrier  is  turned 
T  still  further  until  AC  gives  a  signal, 
FlG-  221-  that  is,  until  RN  bisects  ZFat  OH, 

This  gives  TL9  NHt  and  HL=  y2  (180°  -  TL). 
Then  the  deviation  d  =  LN=  NH  —  HL. 

To  find  an  expression  for  the  index  of  refraction  of  the  substance  : 

Since  i  is  the  critical  angle  of  the  prism,  n  =  n^  sin  i  (p.  90). 

From   the   figure,  a  =  ACB  =  AN,R  =  N^OR  +  N^RO  =  i  +  ft, 

whence  z  =  a  —  /9,  or  in  general  i=  a  =fc  pt  since  as  n  approaches 

«u  ft  diminishes  and  may  become  negative  ;  that  is,  the  ray  RL  may 

be  on  the    other  side  of  RN.     Substituting,  n  =  ^  sin  (a  =t  /9)  = 

;/!  (sin  a  cos  /5  ±  cos  a  sin  /?.)     The  reflected  ray  OR  is  refracted  at  R 

sin  /9       i 

where   -~-  =  -  ,   whence 
sin  d      n 


sin  8  I 

sin  /?  =  -   —  and  cos  /?  =  -  \/// 2  —  sin2 


Substituting, 


sin  a  \/n  2  — sin2<5  ±  cos  a  sin 


/«  ///^  Pulfrich  apparatus  the  glass  prism  is  replaced  by  a  vertical 
glass  cylinder,  the  substance  resting  upon  the  upper  base  and  il- 
luminated from  below  by  diffused  light,  no  light  being  permitted 
to  enter  at  the  top.  The  angle  d  of  deviation  from  the  normal 
on  emergence  is  measured  by  a  right-angled  telescope  revolving 


OPTICAL   CHARACTERS.  95 

on  a  horizontal  axis.     For  this 


n  =-v2— 


In  the  Abbe  apparatus  the  glass  prism  is  replaced  by  a  hemi- 
sphere of  glass  with  the  substance  resting  upon  the  horizontal 
base.  The  critical  angle  is  measured  directly  by  a  telescope 
centred  upon  the  centre  of  the  sphere.  For  this  n  —  ;/x  sin  i. 

The  refracting  liquid  used  may  be  :  Thoulet  solution,  Rohrbach 
solution,  «  mono-Bromnapthalin,  Methylene  iodide  alone  or  satu- 
rated with  iodoform  or  sulphur  or  any  other  sufficiently  stable  and 
transparent  liquid,  the  index  of  refraction  of  which  is  higher  than 
that  of  the  mineral  to  be  tested. 

Density.        Temp.    Li  or  B.   Na  =  D.     Tl  or  E.   Decrease  for  i°  increase, 
Glycerine. 

C3H8O3  1.2594         20  i.47293 

Thoulet  Solution. 

Potassium.  3.122  18     B  1.6960  1.7167    E  1.7391 

Mercury,  2.493  '8     B  1.5855  1.6001    E  1.6160 

Iodide.  2.091  18     B  1.5129  1.5235    E  1.5347 

Rohrbach  Solution. 

Barium.  3  564  23  I-7931    E  1.8265 

Mercury. 

Iodide. 
Bromnapthalin  a. 

C10H7Br.  8  1.66264  .00045  Na 

1.4914         20  1.65820 

Methylene  Iodide* 

CH2T2  8     Li  1.746  1.7466     Tl  1.7584  .00067  Li.  0007  1  Na 

19  1.7421  .00073X1 

II.     OPTICALLY    ISOTROPIC      CRYSTALS     WHICH     ARE     CIRCU- 
LARLY   POLARIZING. 

Certain  isometric  crystals  possess  the  power  of  rotating  the 
plane  of  polarization  (see  p.  100)  of  the  incident  light  ivhatever 
the  direction  of  transmission  ;  they  are  therefore  still  isotropic,  but 
doubly  refracting,  this  rotation  having  been  experimentally  proved 
to  be  due  to  two  rays  transmitted  with  different  velocities  and 
circularly  polarized  in  opposite  directions. 
RAY  SURFACE. 

Either  circularly  polarized  ray  is  transmitted  with  a  constant 
velocity  in  any  direction,  but  with  respect  to  each  other  the 
velocities  have  a  constant  difference,  hence  the  ray  surface  must 


*  Saturation  with  iodoform  raises  the  index  about  .02  and  with  sulphur  as  much  as 
.04.     Li  line  is  at  32;  B  at  28.     Tl  line  is  at  68;  E  at  71. 


96  CHARACTERS  OF  CRYSTALS. 

be  two  concentric  spheres.  Since  on  reversing  any  section  the  di- 
rection of  observed  rotation  is  not  changed  it  follows  that  at  the 
extremities  of  any  diameter  of  the  ray  surface  the  rotations  in  the 
same  shell  must  be  opposite  in  direction.  There  can,  therefore, 
be  no  planes  of  general  symmetry,  though  every  diameter  is  an 
axis  of  isotropy. 

The  division  is,  therefore,  necessarily  limited  to  classes  28  and 
29  which  have  no  planes  of  symmetry.  Examples  in  class  28  are 
barium  nitrate,  sodium  chlorate  and  sodium  bromate.  The 
phenomenon  has  not  yet  been  observed  in  class  29  and  it  is  evi- 
dent that  symmetry  is  not  the  only  determining  cause. 

The  phenomena  and  testing  of  circularly  polarizing  crystals  will 
be  described  more  fully  under  uniaxial  division  IV. 

ABSORPTION  IN  ISOTROPIC  CRYSTALS. 

In  optically  isotropic  crystals  monochromatic  light  diminishes 
steadily  in  intensity  as  the  distance  traversed  increases,  but  is  in- 
dependent of  the  direction  of  transmission. 

With  white  light  the  different  component  colors  are  absorbed  at 
different  rates. 

A  section  of  any  given  thickness  therefore  of  an  isometric 
crystal*  will  transmit  the  same  color  tint  whatever  the  direction 
in  which  the  crystal  may  be  cut. 

By  decomposing  this  color  with  a  prism  the  absorption  spec- 
trum is  obtained,  which  usually  shows  a  gradual  change  in  absorp- 
tion in  adjoining  portions,  perhaps  increasing  from  one  end 
towards  the  other,  perhaps  increasing  in  both  directions  from  the 
centre.  With  even  a  moderate  thickness  certain  colors  may  be 
absorbed  completely  so  that  the  spectrum  shows  dark  bands. 

*The  color  tints  due  to  the  combination  of  the  partially  and  unequally  absorbed  rays 
may  vary  greatly  in  specimens  of  the  same  substance,  which  may  be  properly  colorless  or 
faintly  colored  in  ordinary  thicknesses  and  yet  frequently  occur  of  brilliant  colors,  which, 
nevertheless,  conform  perfectly  in  absorption  to  the  crystal  symmetry.  It  is  prob- 
able that  this  is  due  to  minute  amounts  of  oxides  of  rarer  metals  titanium,  zirconium, 
cerium,  etc.,  dissolved  like  coloring  matter  in  solution.  See  Weinschenk,  Zcit.f.  Anorg. 
Chemie,  XII.,  372. 


OPTICAL   CHARACTERS. 


97 


CHAPTER  VIII. 


THE    OPTICALLY   UNIAXIAL    CRYSTALS. 


In  every  crystal  of  the  hexagonal  or  tetragonal  system  the  direc- 
tions equally  inclined  to  the  crystallographic  axis  c  are  optically 
equivalent,  so  that  c  is  an  axis  of  isotropy  and  being  a  fixed 
crystallographic  direction  may  be  called  an  Optic  Axis.*  ,  All 
diameters  normal  to  c  are  axes  of  binary  symmetry. 

III.     OPTICALLY    UNIAXIAL    CRYSTALS    IN    WHICH    THE    OPTIC 
AXIS  IS  A  DIRECTION  OF  SINGLE  REFRACTION. 

DOUBLE  REFRACTION. 

In  a  moderately  thick  calcitef  cleavage,  Fig.  222,  mounted  with  a 
rhombic  face  vertical  and  so  that  it  can  be  revolved  about  a  hor- 


FIG.  223. 

izontal  axis  normal  to  a  vertical  face,  any  light  ray,  IT,  nor- 
mally incident,  Fig.  223,  at  the  vertical  face,  is  transmitted  in  the 
rhomb  as  two  rays  of  essentially  equal  brightness;}:  (giving  twa 
images  of  any  signal),  and  as  the  rhomb  is  turned  about  the  axis 

*  It  will  be  seen  later  that  while  the  optic  axis  in  uniaxial  crystals  is  fixed,  the  so- 
called  optic  axes  in  biaxial  crystals  change  with  the  light  or  by  heat  or  pressure. 

f  Calcite  is  chosen  because  of  the  marked  divergence  of  the  two  rays.     The  discus- 
sions, however,  are  general. 

\  Absorption  is  more  marked  in  the  case  of  one  image  than  the  other. 


98 


CHARACTERS  OF  CRYSTALS, 


one  of  these  remains  fixed  in  position,  the  other  moves  around  the 
first  and  always  so  that  both  remain  in  a  plane  parallel  to  a  b  c  d 
(the  so-called  principal  section)  and  at  a  constant  distance  apart. 

With  crystals  optically  isotropic  and  normal  incidence  the  fixed 
image  only  would  have  been  seen,  hence  this  is  called  the  ordinary 
and  the  other  by  contrast  the  extraordinary. 

If  a  second  calcite  rhomb  similarly  mounted  is  placed  in  front  of 
the  first  and  revolved,  the  other  remaining  stationary,  each  of  the 
two  rays  from  the  first  is  again  split  into  two  rays,  an  ordinary 
and  an  extraordinary,  lying  in  the  principal  section  of  the  sec- 
ond calcite,  and  these  are  no  longer  of  equal  brightness,  but  wax 
and  wane  in  turn,  the  sum  of  their  intensities  remaining  constant. 
PLANE  OF  VIBRATION. 

The  changes  in  intensity  (brightness)  corresponding  to  different 

values  of  «,  the  angle  between  the 
principal  sections,  correspond  exactly 
to  the  assumption  that  the  varying 
elliptical  vibrations  of  common  light 
are  converted  by > the  first  calcite  into 
two  sets  of  straight-lined  vibrations, 
one  parallel  to  the  principal  section, 
one  at  right  angles  thereto.  Since 
the  rays  are  of  equal  intensity,  with 
equal  vibration  amplitudes,  if  we  de- 
note the  ordinary  and  extraordi- 
nary rays  from  the  first  calcite  by  O 

and  E  and  their  ordinary  and  extraordinary  components  in  the 
second  calcite  by  00  0€  and  E  Ee,  it  will  be  seen  from  Fig.  224. 


If  the  principal  section  is  the  plane 
of  vibration  of  the  ORDI- 
NARY RAY. 

Ray. 

Direction. 

Amplitude. 

0 
E 

ab 

Ok  =  cos  a 
Og  =  sin  a 
Oc=l 

£ 

ai^i 

O&r=s\n  a 

£e 

<A 

Of=  cos  a 

O                cd 

Oc=i 

If  the  principal  section  is  the  plane 
of  vibration  of  the  EXTRA- 
ORDINARY RAY. 

1 

c\d\ 
*& 

O/=  cos  a 
Ok  =  sin  a 

Og  —  sin  a 
Oh  =  cos  a 

OPTICAL   CHARACTERS. 


99 


Both  assumptions  give,  therefore,  the  same  amplitudes  for  the 
four  rays,  viz :  O0  =  cos  a,  Oe  =  sin  «,  £Q  =  sin  a,  Et  =  cos  a, 
therefore  both  correspond  to  the  same  relative  intensites  (propor- 
tionate to  squares  of  amplitudes),  moreover  O0  —  £e  and  Oe  =  EQ 
for  all  values  of  «,  and  also  00  -f Oe  =  O  and  EQ  -f  Ee  =  E  since 
sin2  «-f  cos2  a.  =  i. 

We  shall  hereafter  assume*  //&?/  the  plane  of  vibration  of  the  extra- 
ordinary ray  is  parallel  to  the  principal  section,  and  that  of  the  ordinary 
is  at  right  angles  to  the  principal  section. 


Oo0 


t>. 

»E0                \< 

) 

:                 <S 

e  «•    \ 

&''® 

F.C*OO      % 

o( 

s 

I           * 

'».      :      »• 

; 

FIG.  225. 

The  results*  corresponding  to  different  values  for  a  are  illus- 
trated in  Fig.  225,  a  b  representing  the  principal  section  of  the 
first  calcite  al  bl  that  of  the  second  calcite  and  the  diameter  in  each 
circle  being  the  assumed  vibration  direction.  The  intensities  cor- 
responding are  proportionate  to  the  squares  of  the  vibration  ampli- 
tudes. 


Intensity  proportionate  to  cos2  a 


00 

Ee 

Oe 

E0 

0 

i 

i 

0 

0 

45° 

X 

X 

X 

X 

900 

0 

0 

I 

I 

igoO 

i 

i 

0 

0 

2250 

X 

X 

X 

2700 

0 

0 

I 

I 

600 

X 

X 

§ 

X. 

Intensity  proportionate  to  sin2  a 


That  is  at  zero  two  rays  are  extinguished  and  the  same  two  at  1 80°. 
From  these  points  they  gradually  increase  at  the  expense  of  the 

*  This  supposition  is  purely  for  convenience  as  best  connecting  this  work  with  the 
common  usage  in  "Optical  Mineralogy  and  Petrography."  The  question  is  one  for 
the  physicists  and  by  them  seems  to  be  more  generally  decided  in  the  other  way.  See 
Jas.  MacCullagh,  Trans.  Royal  Irish  Soc.,  XVIII.,  XXI.,  W.  H.  C.  Bartlett,  Amer. 
Jour.  Science,  Nov.,  1890.  F.  Neumann,  Vorlesungen  fiber  der  festen  Korper  una 
des  Lichtdthers. 


ioo  CHARACTERS  OF  CRYSTALS. 

other  pair  and  are  at  a  maximum  at  90°  and  270°,  the  others  being 
then  totally  extinguished.  At  all  diagonal  positions  there  are  visible 
four  rays  of  equal  intensity,  and  for  all  other  positions  as  at  60° 
the  intensity  of  two  rays  are  greater  than  those  of  the  other 
two. 

PLANE  OF  POLARIZATION. 

Common  light  reflected  at  a  particular  angle  of  incidence  char- 
acteristic of  the  reflecting  substance  acquires  the  same  peculiar 
characters  as  the  rays  produced  by  double  refraction. 

If  normally  incident  at  a  rhombic  face  of  a  calcite  rhomb  an  or- 
dinary (unrefracted)  image  is  obtained  when  the  plane  of  reflection 
(through  incident  and  reflected  ray),  is  parallel  to  the  principal  sec- 
tion of  the  calcite,  an  extraordinary  when  these  are  at  right  angles 
to  each  other  and,  for  all  other  angles,  both  ordinary  and  extra- 
ordinary of  varying  intensity,  just  as  with  calcite. 

Malus*  described  the  reflected  ray  as  POLARIZED  with  reference 
to  the  plane  of  reflection  and  called  the  latter  the  PLANE  OF  POLAR- 
IZATION of  the  ray. 

In  the  same  sense  the  two  rays  produced  from  common  light 
by  double  refraction  in  calcite  are  said  to  be  POLARIZED. 

The  principal  section  of  the  analyzing  calcite  becomes  the  plane 
of  reference.  Whatever  it  is  parallel  to  wlien  a  ray  undergoes  ordi- 
nary refraction  is  the  plane  of  polarization  of  that  ray\  that  is,  the 
plane  of  polarization  of  the  ordinary  ray  is  the  principal  section  and 
the  plane  of  polarization  of  the  extraordinary  ray  is  at  right  angles 
to  the  principal  section. 

RAY  SURFACE. 

By  measurement  of  the  indices  of  refraction  for  different  direc- 
tions of  transmission  it  is  found  : 

1°.  That  the  velocity  of  the  ordinary  ray  is  constant. 

2°.  That  the  velocity  of  the  extraordinary  ray  in  any  section 
through  c  varies  for  different  directions  of  transmission.  For  the 
direction  parallel  to  c  it  is  equal  to  that  of  the  ordinary  and  dif- 
fers most  for  the  direction  of  right  angles  to  c  andf  or  any  other 
direction,  as  discovered  by  Huyghens  for  calcite,  the  velocity  is 
given  by  the  corresponding  radius-vector  of  an  ellipse  the  axes  of 
which  are  the  least  and  greatest  velocities. 

The  ray  surface  for  light  of  any  wave-length  is  therefore  a  double 
surface,  the  extraordinary  shell  being  an  ellipsoid  formed  by  the 

*Mem.  de  la  Soc.  de  Strasbourg,  1811,  I.,  284. 


OPTICAL   CHARACTERS. 


101 


revolution  of  the  ellipse,  the  axes  of  which  are  the  least  and  greatest 
velocities,  about  one  of  its  axes  ;  and  the  ordinary  shell  a  sphere 
with  a  diameter  equal  to  the  axis  of  revolution  of  the  extraor- 
dinary. This  surface  is  symmetrical  to  all  planes  through  the  optic 
axis  and  to  the  diametral  plane  at  right  angles  thereto. 

Denoting  the  indices  of  refraction  of  the  fastest  and  slowest 
rays,  that  is  of  the  two  rays  transmitted  normal  to  the  optic 
axis,  by  «  and  y  and  their  vibration  directions  by  a  and  c,* 
there  will  be  two  cases  arbitrarily  distinguished  as  POSITIVE  in 
which  c  is  parallel  to  the  crystallographic  axis  c  and  NEGATIVE  in 
which  a  is  parallel  to  c.  In  the  former  case,  therefore,  the  extra- 
ordinary ray  with  vibration  direction  parallel  c  will  be  the  slower 
ray  and  the  extraordinary  shell  will  be  wholly  within  the 
ordinary  as  in  Fig.  226,  while  in  the  negative  surface  the 
extraordinary  ray  is  the  faster  and  the  extraordinary  shell  will 
enclose  the  ordinary  as  in  Fig.  227. 
THE  OPTICAL  INDICATRIX. 

All  the  relations  between  the  optical  characters  of  a  crystal  can 
be  expressed  by  the  geometrical  characters  of  the  extraordinary 
shell  or  its  equivalent/)*  the  ellipsoid  of  revolution,  the  axis  of  revo- 
lution of  which  is  the  index  of  refraction  of  the  extraordinary  ray 
transmitted  normal  to  c  and  the  equatorial  diameter  the  index  of 
the  ordinary  ray. 


FIG.  226,  -f ,  c  —  c.  FIG.  227,  — ,  c=  a. 

In  Fig.  226  let  the  section  of  the  extraordinary  shell  represent 
the  corresponding  section  of  the  indicatrix,  then  let  IO  be  any  di- 
ection  of  transmission.  Draw  ID  tangent  at  /,  OE  parallel  to 

*  Often  called  axes  of  elasticity. 

f  In  indicatrix  axis  in  direction  a  —  a,  in  shell  =  —  •     In  indicatrix  axis  in  direction 

7 

c  =  7,  in  shell  =  —  •    But  a  :  y  =  —  :  —  • 


102  CHARACTERS  OF  CRYSTALS. 

ID,  ED  tangent  at  E  and  EN  normal  to  JO,  then  are  01  and  OE 
conjugate  radii  and  the  parallelgram  OEDI  is  of  constant  area 
Oc  x  Oa,  hence,  Oc  x  Oa  =  01 X  £7V.  Let  OA  denote  the  nor- 
mal at  0  equal  Oa> 

Describe  the  circumscribing  circle  or  section  of  the  ordinary 
shell,  then  are  Of  and  OH  the  velocities  of  extraordinary  and  or- 
dinary ray  ve  and  VQ. 

From  Oc  X  Oa  =  Of  x  EN,  have  01=  v^  = -,., 

JH.V 

Ocx  Oa 


From  <9#  =  6>c  and  OA  =  Oa,  0/f  =  z>0  = 


OA 
that  is  v*  :  vn  = 


Moreover,  jSTV  in  the  principal  section  and  OA  normal  to  it  are 
respectively  the  directions  of  vibration  of  rays  to  which  they  cor- 
respond. 

That  is :  For  any  diameter  of  the  ellipsoid  considered  as  a  direc- 
tion of  transmission  there  are  two  points  of  the  surface,  the  nor- 
mals from  which  are  also  normal  to  that  diameter.  These  normals  are 
at  once  the  directions  of  vibration  and  the  reciprocals  of  the  velocities 
of  the  rays  transmitted  in  the  direction  of  the  diameter. 

DERIVATION  OF  POSITIVE  RAY  SURFACE. — Upon  a  and  c,  Fig.  226, 
the  directions  of  vibration  of  the  fastest  and  slowest  rays,  make 
Oa  =  a  Oc  =  Y-  Oc  is  the  axis  of  rotation  and  the  ellipsoid  result- 
ing is  the  indicatrix. 

Section  ac  of  Ray  Surface. — For  the  direction  Oc  the  two  normals 
are  Oa  and  Oa,  for  the  direction  Oa  the  two  normals  are  Oa  and  Oc, 
for  any  other  direction,  Of,  the  two  normals  are  OA  =  Oa  and 
EN,  in  which  EN  varies  between  Oa  and  Oc  according  to  the  di- 
rection of  transmission. 

According  to  the  rule  then  this  section  of  one  shell  is  a  circle 

with  constant  radius  -^  =-  and  of  the  other  shell  is  an  ellipse 
Oa     a 

with  axis  in  direction  a  =  7^-  =  •  -  and  in  direction  c  =  7^-  =  - 

Oc       r  Oa        a 

corresponding  exactly  to  Fig.  226. 

SECTION  cm- — For  every  direction  of  transmission  two  normals 
exist,  Oc  and  Oa ;  that  is,  this  section  of  the  double  surface  is  two 

I        i        .    i         I 

concentric  circles  with  radii  -=-  =  —  and  -=-=—. 

Oc       r         Oa       a 


OPTICAL   CHARACTERS. 


103 


Determination  of  Optical  Characters. 

The  determination  of  the  optical  characters  of  a  uniaxial  crys- 
tal consists  essentially  in  the  determination  of  the  axes  of  the  Indi- 
cate ix ;  that  is,  of  the  principal  indices  of  refraction  (indices  of  the 
two  rays  transmitted  normal  to  the  optic  axis  c\  The  determina- 
tion may  be  direct  or  indirect. 

Direct  Determination  of  Principal  Indices  of  Refraction. 

(a)  Prisms  with  refracting  edge  Bt  Fig.  212,  parallel  to  the  optic 
axis  give  for  minimum  deviation,  p.  88,  a  direction  of  transmis- 
sion RS  normal  to  the  optic  axis. 

(b)  In  prisms  with  refracting  edge  B,  Fig.  212,  perpendicular  to 
the  optic  axis  and  faces  AB  and  AC  equally  inclined  thereto  the 
optic  axis  is  BD  and  the  direction  of  transmission  RS  for  minimum 
deviation  is  normal  to  it. 

Because  the  horizontal  section  aa  of  the  ray  surface  is  two  con- 
centric circles,  the  formula  p.  89  holds  good  for  both  rays 

sin  y2(§  +  x]  sin  V2  ($'  -f  x\ 

a  —  n,  — .    v  .  r—n\ .      , 

sin  y2  x  sin  y2  x 

(c]  In  a  prism  ABC,  Fig.  228,  with  one  face 
AB  parallel  to  the  optic  axis,  rays  normally  in- 
A  cident*  at  that  face  experience  no  refraction, 
because  in  sections  normal  to  the  optic  axis 
both  shells  are  circles,  but  on  emergence  from 
the  second  face  the  two  rays  are  differently  re- 
fracted. 

Denoting  the  prism  angle  by  x,  the  deviation 
of  the  faster  ray  by  d  and  that  of  the  slower  ray 
by  Sr  and  measuring  these  only  we  have : 


a  = 

sin  PSN 

w                      — 

M  sin  (x  4-  d) 

'Sin  737V- 

sin  OSN 
Sin  737V  ~ 

t  sin  (x  4  8') 

1        sin;r 

*  With  this  face  vertical  and  with  collimator  and  telescope  at  any  convenient  angle 
obtain  a  signal  from  the  face,  then  turn  the  crystal  through  one-half  this  angle  until 
normal  to  the  collimator. 


104  CHARACTERS  OF  CRYSTALS. 

(d)  The  indices  «  and  f  may  be  calculated  from  the  indices  ob- 
tained with  other  prisms.* 

(e)  In  any  crystal  face  or  section  one  of  the  extinction  direc- 
tions, p.  117,  is  in  a  plane  through  the  optic  axis  and  the  other  is 
at  right  angles  thereto  and,  therefore,  is  itself  at  right  angles  to  the 
optic  axis. 

If  this  direction  is  made  horizontal  in  a  total 
reflectometer  the  transmission  will  be  at  right 
angles  to  the  optic  axis  and  the  methods  and 
formulae  of  pp.  90-95  will  be  available.  There 
will  be  two  distinct  limit  lines  which  may  both 
be  in  the  field  at  once  as  in  Fig.  229  or  may 
not,  and  are  successively  brought  into  coinci-  FIG.  229. 

dence.with  the  vertical  hair  of  the  telescope. 

The  measurements  determine  the  relative  values  of  «  and  ?-.  By 
means  of  a  nicols  prism,  p.  105,  which  transmits  only  light  vibra- 
ting in  a  plane  through  its  shorter  diagonal,  the  ordinary  and  extra- 
ordinary rays  may  be  distinguished,  the  former  being  transmitted 
when  the  shorter  diagonal  is  at  right  angles  to  the  optic  axis,  the 
latter  when  the  shorter  diagonal  is  parallel,  as  previously  explained. 

In  positive  crystals  the  ordinary  is  the  faster,  that  is,  corresponds 
to  «. 

In  negative  crystals  the  ordinary  is  the  slower,  that  is,  corresponds 
to  Y- 

Indirect  Determination  with  Plane  Polarized  Light. 

Parallel  faced  (plane-parallel)  sections  of  known  orientation  are 
prepared.  Cleavages  are  used  when  obtainable,  or  if  the  section 
is  to  be  parallel  to  a  crystal  face  this  face  is  cemented  to  a  glass  and 
an  opposite  artificial  face  ground  on  with  emery  and  polished  with 
rouge.  When  the  desired  section  is  not  parallel  to  any  known 
face  it  is  fastened  to  glass  by  slowly  hardening  cement,  adjusted  at 
the  proper  angle  and  ground,  the  new  face  being  verified  gonio- 
metrically  with  reference  to  other  faces. 

Crystals  soluble  in  water  are  ground  in  some  other  liquid,  as 
mono-bromnapthalin  or  benzine,  and  if  fragile  are  ground  only  on 
a  glass  plate.  After  grinding  the  sections  are  cleaned  and  trans, 
ferred  to  another  plate. 

A  very  perfect  apparatus  in  which  true  planes  may  be  rapidly 

*Th.  Liebisch,  Phys.  Kryst.,  1891,  384-390. 


OPTICAL   CHARACTERS. 


105 


ground  within  10'  of  any  desired  direction  has  been  described  by 
A.E.  Tutton.* 

PLANE  POLARIZED  LIGHT  may  be  produced  from  common  light, 

(a]  By  reflection  at  a  particular  angle  of  incidence  (tan  /=  »), 

the   vibrations   being  at   right   angles   to  the  plane  of  reflection 

(plane  through  incident  and  reflected  ray)  in  accordance  with  the 

assumption  of  p.  99. 

(&}  By  refraction  through  a  series  of  parallel  glass  plates,  each 
plate  increasing  the  proportion  of  polarized  light.  In  this  case  the 
vibrations  are  in  the  plane  of  reflection. 

In  (a]  and  (b]  common  light  is  always  present. 
(c]  By  double  refraction  and  total  reflection  of  one  of  the  rays. 
The   best  known  device  for   securing  this  effect  is  the  so-called 
Nicol's  prism,t  made  from  a  cleavage  of  calcite  with  a  length  about 
twice   its   thickness,  Fig.  230.     The   two  small 
rhombic   faces  at  71°  to  the    edge  are   ground 
away  and  replaced  by  faces  at  68°  to  the  edge. 
The   prism  is  then  cut  through  by  a  plane  at 
right  angles  both  to  the  new  terminal  faces  and 
to  the  principal  section.     The  parts  are  carefully 
polished  and  cemented  by  Canada   balsam,  the 
index  of  refraction  of  which  is  1.54  or  about  that 
of  the  extraordinary   ray   bd^  which,  therefore, 
passes  through  the  balsam  with  but  little  change 
in  direction;  the  ordinary  ray  be,  however,  with 
an  index  of  refraction  of  1.658,  being  incident  at 
an  angle  greater  than  its  critical  angle,  is  totally 
reflected.  The  vibration  direction  of  the  emerging 
light  is,  therefore,  parallel  to  the  short  diagonal 
of  the  face  of  the  nicol,  as  shown  by  the  arrow. 
(d)  By  double  refraction  and  absortion.     Cer- 
tain  substances    absorb    one    ray   much    more 
FIG  2  o  rapidly  than  the  other,  hence  thicknesses  can  be 

*  Proc.  Royal  Soc.     1894.     Vol.  55,  p.  108. 

t  Described  Jamesons  New  Journal,  V.  6  1828.  Various  modifications  of  this 
prism  have  been  made  to  decrease  the  cost  and  increase  the  field.  See  Ze.it.  f.  Kryst., 
XL,  179,  410,  for  instance. 

The  Foucault's  prism  uses  a  layer  of  air  instead  of  Canada  balsam  and  is  cut  at  a 
different  angle  requiring  a  shorter  prism  but  giving  a  smaller  field. 

In  the  Hartnack  prism  the  terminal  planes  are  at  right  angles  to  the  axis,  the  prism 
is  shorter  and  the  field  reaches  420.  It  is  much  used. 

The  Bertrand  prism  is  of  flint  glass  with  the  high  index  of  refraction  of  1.658.  It 
is  bisected  by  a  plane  at  76°  43'  to  the  base  and  between  the  two  halves  is  a  thin  calcite 


io6  CHARACTERS  OF  CRYSTALS. 

chosen  for  which  one  is  totally  absorbed,  the  other  is  partially 
transmitted  as  light,  the  vibrations  of  which  are  in  one  plane.  In 
tourmaline  the  ordinary  ray  is  the  more  rapidly  absorbed. 

INTERFERENCE.  —  A  ray  of  polarized  monochromatic  light  AB 
Fig.  231,  incident  at  the  lower  surface  of  a  plane-parallel  doubly 
refracting  plate  at  any  angle,  is  broken  into  two  rays  BC  and  BDY 
vibrating   in  planes  at  right  angles  to  each 
III  other   and    following    different   paths   in   the 

1   CJ  £>]  plate.     On  emergence  they  follow  parallel  but 

not  coincident  paths  and  do  not  produce  inter- 
ference. 

But  among  the  other  incident  rays  from  the 
same   source   and   parallel   to  AB   there  are 


rays  EG  and  FH,  such   that  from  all  points 


231.  £  ancj  £  o^  ^  Upper  surface  there  will  emerge 

the  ordinary  component  of  one  ray  and  the  extraordinary  of 
another  following  the  same  path.  These  rays  will  have  travelled 
over  slightly  different  paths  in  the  plate  with  different  velocities. 

If  a  second  polarizer  is  placed  in  the  path  of  these  rays  each  ray 
will  be  by  it  resolved  into  components  the  vibrations  of  which  are 
in  and  at  right  angles  to  the  plane  of  vibration  of  the  polarizer 
and  only  the  former  will  be  transmitted.  That  is,  there  will 
emerge  two  rays  advancing  in  the  same  line  and  with  parallel  vi- 
brations. If  these  vibrations  are  alike  in  phase  the  intensity  of  the 
resultant  ray  will  be  proportionate  to  the  square  of  the  SUM  of  their 
amplitudes,  but  if  unlike  in  phase  the  intensity  will  be  propor- 
tionate to  the  square  of  their  DIFFERENCE. 

POLARISCOPES.  —  The  instruments  used  for  producing  and  study- 
ing the  interference  phenomena  are  called  POLAHISCOPES.  In  these 
parallel  rays  of  plane  polarized  light,  or  converging  bundles  of  par- 
allel rays,  are  incident  at  one  surface  of  the  plate  at  a  known 
angle,  traverse  the  plate  undergoing  single  or  double  refraction  ac- 
cording to  its  nature;  and,  if  doubly  refracting,  the  rays  following; 
the  same  path  are  reduced  by  the  analyzer  to  one  plane  of  vibra- 
tion producing  interference  phenomena. 

The  essentials  of  a  polariscope  for  parallel  light  are  shown  in 


cleavage  properly  oriented.  The  light  enters  the  prism,  and  reaching  the  calcite  is 
doubly  refracted ;  the  ordinary  ray,  with  a  refractive  index  about  that  of  the  glass,  con- 
tinues its  course;  the  extraordinary  with  a  much  lower  index  is  totally  reflected.  The 
field  is  about  45°.  Compte  Rendu,  Acad.  Sci.,  Sept.  29,  1884. 


OPTICAL   CHARACTERS. 


107 


Fig.  232.  The  mirror  M  sends  parallel  rays  through  the  lower 
lens  L,  which  concentrates  them  at  the  centre  of  the  polarizer  P\ 
this  point  is  also  the  focus  of  the  equivalent  upper  lens  L.  On 
emergence  from  L  the  rays  are  again  parallel,  undergo  refraction 
in  the  plate  5  of  the  substance  and  reach  the  analyzer  A,  which 
transmits  only  those  components  of  the  resultant  rays  the  vibra- 
tions of  which  are  in  its  own  plane. 

Convergent  light  is  obtained  by  the 

S  addition  of  a  lens  or  system  of  lenses 

,,  of  short  focal  length  just  above  the 
plate  5  and  a  corresponding  system 
just  below  the  plate,  Fig.  233.  Any 
point/  of  the  focal  plane  of  the  lower 
lens  system  is  illuminated  by  a  cone 
of  rays  the  base  of  which  is  the  lens. 
This  cone  is  made  a  cylinder  of  par- 
allel rays  by  the  lens.  The  rays  of 
each  cylinder  which  traverse  the  plate 
are  again  concentrated  by  the  upper 
lens  system  at  points  />',  etc.,  are 
sorted  by  the  analyzer  and  finally 
exhibit  a  picture  or  image  the  shape 
brightness  and  tints  of  which  depend 
upon  the  structure  of  the  plate  for  all 
the  directions  traversed  by  the  cylin- 
ders of  parallel  rays. 

The  polariscope  of  to-day  is  usually  a  polarizing  microscope. 
In  the  simpler  types,  such  as  the  Seibert  *  n  A,  the  polarizer  below 
the  stage  can  be  raised,  lowered  and  turned ;  the  analyzer  above 
the  objective  can  be  pushed  in  and  out  and  convergent  light 
images  can  be  obtained  with  high  power  objective  by  placing  a 
small  convergent  lens  on  top  of  the  polarizer,  raising  the  latter  till 
it  touches  the  section  and  removing  the  eye-piece.  An  orifice 
above  the  objective  is  always  provided  for  the  insertion  of  test  plates, 
p.  146,  and  with  increasing  complexity  there  are  added  special 
micrometer  eye-pieces,  Bertrand  lens  for  magnification  of  con- 
vergent light  image,  sliding  motions  of  the  stage  and  so  on. 


_ 
P\ 


232. 


FIG.  233. 


*  For  description  of  this  instrument   and   its   manipulation,   see    L.  Mel.  Luquer, 
S.  of  M.  Quarterly,  1896,  p.  442-445. 


io8 


CHARACTERS  OF  CRYSTALS. 


FIG.  234. 

The  Fuess  microscope  *  model  VI.,  Fig.  234,13  at  present  prob- 
ably the  finest  instrument  made  for  this  work.  The  stage  reads  to 
minutes  and  has  quick  rotation  by  hand,  slow  rotation  by  ratchet 
and  sliding  motions  in  two  directions.  There  is  an  independent 


*  Aeues  Jahrbuch  f.  Mineralogie  Beilage,  Bd.,  X.,  180,  by  C.  Leiss. 


OPTICAL   CHARACTERS. 


109 


FIG.  236. 

focussing  screw  for  the  interference  figure  and  a  special  device  of 
cog  wheels  r  Z,  rl  Zl,  by  which  there  may  be  a  simultaneous 
rotation  of  polarizer  and  a  special  cap  analyzer,  the  object  remain- 
ing at  rest,  but  the  same  relative  change  taking  place  as  if  the  stage 
were  revolved  and  the  nicols  at  rest. 

Two  forms  of  the   Norremberg   apparatus    as  constructed  by 
Fuess   are  here  shown.     Fig.  235   shows  the  so-called  Universal 


no 


CHARACTERS  OF  CRYSTALS. 


Apparatus,  ee'  are  collecting  lenses  on  each  side  of  the  polarizer ; 
above  e'  are  four  plano-convex  lenses,  n,  forming  the  condenser 
and  just  over  these  the  stage. 

In  a  separate  tube  system  above  are  the  objective,  composed  of 
four  similar  plano-convex  lenses  o,  and  at  their  focal  plane  the 
glass  plate  r,  on  which  a  cross  and  a  scale  are  marked ;  the  image 
there  formed  is  magnified  by  t  and  viewed  through  the  analyzer  q. 

By  removal  of  n,  o,  r  and  /  the  apparatus  yields  parallel  rays. 

Fig.  236  shows  a  later  less  expensive  type,  in  which  the  lower 
nicol  is  replaced  by  a  pair  of  mirrors.  The  high  cost  of  iceland 
spar  is  the  principal  reason  for  the  change  and  the  results  are  very 
satisfactory. 

WITH  PARALLEL  MONOCHROMATIC  LIGHT,  AND  CROSSED  NICOLS. 

With  crossed  nicols  none  of  the  light  from  the  polarizer  can 
pass  through  the  analyzer  and  the  field  must  be  dark. 


FIG.  237. 


FIG.  238. 


In  sections  normal  to  the  optic  axis  the  field  remains  dark  through- 
out the  entire  rotation  of  the  stage,  and  no  interference  phenomena 
are  possible,  because  the  light  from  the  polarizer  traverses  the 
section  in  the  direction  of  the  optic  axis,  therefore,  without  change. 

In  all  other  sections  there  is  double  refraction  and  interference. 
The  field  is  dark  at  intervals  of  90°  ;  that  is,  whenever  the  planes 
of  vibration  of  the  rays  produced  in  the  section  coincide  with 
the  planes  of  vibration  of  the  nicols.  For  all  other  positions  the 
field  is  illuminated  by  the  components  of  the  rays  which  pene- 
trate the  analyzer  and  this  brightening  is  most  intense  in  the  diag- 
onal positions. 


OPTICAL   CHARACTERS.  in 

The  rays  pursuing  the  same  path  are  by  the  analyzer  brought 
into  one  plane  of  vibration  and  there  interfere,  the  kind  of  inter- 
ference being  determined  by  J,  the  difference  in  the  retardations 
which  the  two  rays  have  undergone,  the  formula  being* 


Or  for  normal  incidence, 

A  =  /(«,  -  n) 

In  which  J  is  the  retardation  (difference  in  retardation)  in  ////. 
millionths  of  a  millimeter. 

/  is  the  thickness  of  the  plate  in  ////. 

A    is  the  wave-length  in  /J./J.. 

i    is  the  angle  of  incidence. 

i?l  is  the  index  of  refraction  of  the  slower  ray. 

;/  is  the  index  of  refraction  of  the  faster  ray. 

When  J  =  A,  2A,  3 A,  etc.,  the  field  ts  dark  during  an  entire  revolu- 
tion,tor  Fig.  237,  the  components  of  PP,on  emergence  from  the 
plate  with  vibration  directions  RR  and  DD,  must  be  of  the  same 
phase  that  is  the  simultaneously  displacing  forces'  acting  upon  any 
ether  particle  0  are  Or  and  Os,  which  when  reduced  to  the  plane 
of  the  analyzer  are  Oa  and  Oalt  in  opposite  directions  and  equal. 

When  J  =  1A,  fA,  fA,  etc.,  the  light  will 
be  at  its  brightest  because  the  components 
of  PP  must  then  on  emergence  from  the 
plate  be  of  opposite  phase,  Fig.  238,  and 
the  simultaneous  displacing  forces  acting 
on  any  ether  particle  O  are  Or  and  Os, 

which  reduced  by  the  analyzer  to  its  plane  are  Oa  and  Oal  in  the 
same  direction  and  equal. 
EXPERIMENT. 

If  a  wedge  of  double  refracting  crystal,  Fig.  239,  cut  so  that  its 
planes  of  vibration  are  parallel  to  the  length  and  breadth,  is  placed 
between  crossed  nicols  and  illuminated  by  perpendicularly  incident 
monochromatic  light  and  there  revolved. 

It  will  be  perfectly  dark  when  in  the  normal  positions  and  in  all 
others  will  show  a  series  of  dark  and  light  parallel  stripes  which 
are  most  marked  in  the  diagonal  position.  If  the  nicols  are  made 
parallel  the  portions  formerly  light  become  dark.  With  light  of  a  dif- 
ferent wave-length,  the  distance  between  the  dark  bands  is  changed. 


*  Reduced  from  formula,  p.  364,  Glazebrook's  Physical  Optics. 


1  1  2  CHARACTERS  OF  CR  YSTALS. 

*These  relations  may  be  deduced  from  the  formula  for  intensity 
of  emerging  light 

/=  a2  sin2  2  (.  sin2     — 


in  which 

a  =  amplitude  of  incident  ray,  /^—wave-length,  J  =  retardation, 
•<P  =  angle  between  vibration  plane  of  lower  nicol  and  slower  ray. 
/  will  be  a  minimum  : 

(a)  when  sin2  2  <p  =  o  or  2  ?  =  o°,  1  80°,  360°,  540°,  etc.,  or  <p  = 
0°,  90°,  1  80°,  270°;  that  is,  four  times  in  a  revolution  of  the  plate 
or  whenever  the  planes  of  vibration  of  the  plate   coincide   with 
those  of  polarizer  or  analyzer. 

(b)  when  sin2    (  —  I  =  o  which  will  be  whenever  -  =  I,  2,  3,  etc., 

\    A    /  / 

that  is  whenever  the  phase  difference  J  is  a  multiple  of  A,  for  then 
sin2  I  —  I  becomes  sin2  180°,  or  a  multiple  thereof. 

This  is  independent  of  <p.  hence  with  this  condition  the  plate-  will 
remain  dark  throughout  an  entire  revolution. 
/  will  be  a  maximum: 

When  sin2  2^=1  or  2^  =  90,  270,  etc.,  and  ^  =  45°,  135°* 
225°,  etc. 

When  f  =i,  2",  f,  for  then  sin2  (~  J\  =  sin'2  90°,  sin2  270°,  etc. 

=  I.  This  will  be  independent  of  c?  and  the  illumination  will 
exist  throughout  an  entire  revolution. 

WITH  PARALLEL  WHITE  LIGHT  AND  CROSSED  NICOLS. 

In  sections  normal  to  the  Optic  Axis  the  field  is  dark  throughout 
the  rotation,  the  optic  axis  being  the  same  for  all  colors. 

In  all  other  sections  there  is  extinction  every  90°  and  greatest 
brightness  in  the  diagonal  positions,  but,  since  J  may  be  at  the 
same  time  approximately  an  even  multiple  of  ^/  and  an  odd  mul- 
tiple of  j£/',  light  of  one  wave-length  may  be  greatly  weakened 
while  that  of  another  wave-length  is  practically  undimmed;  that  is, 
there  will  result  a  tint  due  to  unequal  changes  in  all  the  colors. 

As  J  increases  in  value  it  passes  alternately  through  */£,  I,  3/2, 
2,  5/2  times  each  wave-length  as  shown  in  Fig.  240,  in  which  the 

*Th.  Liebisch,  Grundriss  der  Phys.  Kryst.>  1896,  p.  271-275. 


OPTICAL   CHARACTERS. 


Blaci 


/  ron  -gray 


Lavender-  gray 


WhiU 


YelU»o 


Oranye-ytttow .   430 
Orange  45o 


Jnd'igo 


Blue 


Green 


.  yellow  yio 

Orange  94  J 

F 

ft  eddn/i -orange  <oo 


Biuuf,  eioUt 


n  5 
D  H 


.As/    /    /     / 


YY 


/   Y  NX 


x/    \ 


^)O 

\ 


£(£»)• 


zigzag  lines  show  the  changes 
in  brightness  for  six  of  the 
colors.*  The  tint  correspond- 
ing to  any  value  of  A  maybe 
judged  by  noting  which  colors 
are  near  maximum  and  mini- 
mum. 
FIRST  ORDER  COLORS. 

A  =  o.— All  light  is  shut  out; 
as  A  increases  *4/  of  violet  is 
first  reached  and  J^A  of  blue 
and  green  next. 

J  =  100  /jt,  /jt. — The  weakly 
coloring  violet  and  still  fainter 
blue  and  green  give  a  total 
impression  of  lavender  gray, 
which  gradually  brightens  as 
the  other  colors  show  through. 

j  =  259  //,  /Jt. — The  color  is 
pure  white i  after  which  the 
violet  end  begins  to  diminish 
in  intensity  and  the  red  end 
to  increase. 

j  =  300  /A,  //.. — Bright  yel- 
low is  at  a  maximum  and  violet 
nearly  extinguished.  Green 
and  red  are  weakened  and  to- 
gether produce  white,  hence 
the  predominating  color  is 
yellow. 

A  =  450  /Jt,  //. — The  yellow 
is  still  high,  but  the  red  rays 
are  at  or  near  their  maximum 
and  the  other  rays  relatively- 
weak;  the  color  is  therefore 
orange. 

A  =  5  30  nn  — Violet  and  red 


FIG.  240. 
are   about  equally  near    a  maximum,  green  is  extinguished  and 

*  The  values  of  A  for  these  in  ///z  (millionths  of  a  millimeter)  are  /^(violet),  393.3 ; 
/^(blue),  486.0;  ^(green),  526.9;  /^(yellow),  589.5;  C(ved),  656.2;  ^(red) 
760.4. 


ii4  CHARACTERS  OF  CRYSTALS. 

blue  and  yellow  weak,  hence  the  strongly  coloring  red  predomi- 
nates. 
SECOND  ORDER  COLORS. 

J  =  575  fj.fjL. — This  is  3/2  A  for  the  brightest  violet,  A  for  bright- 
est yellow  and  red  and  blue  much  weakened,  hence  the  total  im- 
pression is  violet,  often  called  sensitive  violet  and  used  in  testing  be- 
cause with  very  slight  change  in  A  it  becomes  either  red  or  blue. 

J  =  589  yields  indigo  blue. 

A  =  664  up. — This  is  3/2  A  for  the  brightest  blue  and  is  A  far 
orange  red  and  near  it  for  yellow,  hence  predominating  tint  is  blue. 

A  =  800  fjLfi. — This  is  3/2  for  bright  green  and  near  A  for  outer- 
most red  and  2  A  for  violet,  hence  the  color  is  a  mixture  of  green, 
blue  and  yellow  ;  that  is,  green. 

A  =?  900  AV*. — This  is  3/2  A  for  yellow  and  2  A  for  blue.  Some 
red  and  violet  emerge  with  the  yellow;  that  is,  the  prevailing  color 
is  orange. 

A  —  io60fj.ta. — This  2  A  for  green,  3/2  A  for  some  red,  5/2  A  for 
indigo,  and  as  red  is  the  stronger  color  red  predominates. 
COLORS  OF  HIGHER  ORDERS. 

A  =  1 1 30  pp  yields  sensitive  violet  No.  2. 

With  increasing  values  for  A  the  latter  becomes  an  approxi- 
mately perfect  multiple  of  1/2  A  or  A  for  an  increasing  number  of 
wave-lengths  and  the  colors  resulting  are  less  pure  and  brilliant, 
for  example : 

A  =  1 590  /JL/J.. — This  is  5/2  A  for  orange  red,  7/2  A  for  indigo,  3  A  for 
green,  2  A  for  red,  4  A  for  violet.  The  resultant  total  effect  is  a  red. 

With  the  still  higher  values  this  is  further  noticeable  and  beyond 
the  fourth  order  the  tints  resulting  are  not  to  be  distinguished 
from  white.  Hence  thick  crystals  show  no  interference  colors. 

Experiment.  The  quartz  wedge  with  white  light  will  show  colors 
in  the  order  named,  or  the  Federow  mica  wedge  may  be  used,  or 
assuming  the  value  of  (n±  —  n)  for  special  minerals  and  measur- 
ing /,  the  colors  in  plates  of  different  thicknesses  may  be  com- 
pared. 

The  colors  of  corresponding  thicknesses  of  the  mica  wedge,  in 
which  n\  —  n  =  .042  ///./.  are  much  higher  than  for  quartz  with 
;/x  —  n  =  .oog/j.fj.,  one  fifth  the  thickness  producing  the  same  retar- 
dation in  mica. 

The  thicknesses  corresponding  to  an  interference  color  of  red  of 
first  order,, a// =  .009,  for  several  common  minerals  are  in  millimeters 
approximately. 


UNIVERSITY 

OPTICAL   CHARACTERS. 


Calcite  0.003 
Muscovite  0.013 
Chrysolite  0.015 
Barite  0.048 


Quartz  0.060 
Gypsum  0.06 1 
Orthoclase  0.079 
Apatite  0.124 


WITH  CONVERGENT  LIGHT  AND  CROSSED  NICOLS. 

The  bundles  of  parallel  rays,  p.  107,  each  produce  interference 
phenomena  similar  to  those  described,  but  since  for  oblique  inci- 
dence 


A 


n      n 


the  values  of  J  corresponding  to  odd  or  even  multiples  of  y2  A  will 
depend  not  only  on  the  value  (n±  —  11)  and  the  thickness,  but  upon 
the  angle  of  incidence. 

In  sections  normal  to  the  Optic  Axis  there  will  be  a  dark  cross,  the 
arms  of  which  intersect  in  the  optic  axis  (centre  of  field)  and  re- 
main parallel  to  the  vibration  planes  of  the  nicols  during  rotation 
of  the  stage. 

With  monochromatic  light,  if  the  section  is  not  too  thin,*  the 

optic  axis  will  be  surrounded  by  con- 
centric circles  alternately  dark  and 
light,  Fig.  241 .  If  the  greatest  value 
of  A  is  less  than  A  no  rings  will  show. 
The  distance  apart  of  the  dark 
rings  decreases  as  the  thickness  of 
the  section  increases  and  as  the  dis- 
tance from  the  centre  increases,  for 
both  causes  merging  sooner  or  later 
into  a  uniform  brightness. 

With   white   light   the   rings    be- 
come color  rings  strictly  in  the  order 
of  Newton's  colors  if  the  space  per- 
mits, but  often  overlapping  and  finally  merging  into  essentially  uni- 
form tints. 


*  In  the  accidental  orientation  of  rock  sections  there  is  rarely  found  a  perfect  basal 
section  and  the  thickness  is  frequently  insufficient  to  produce  rings.  The  uniaxial  figure 
is  nevertheless  to  be  recognized  in  not  too  oblique  sections  by  one  or  both  arms  of  the 
straight  armed  cross  which  remain  straight  and  parallel  to  the  original  position  on  rota- 
tion of  stage,  whereas  in  biaxial  not  only  do  the  arms  curve  into  hyperbola,  but  revolve 
in  opposite  directions  to  the  rotation  of  stage. 


n6  CHARACTERS  OF  CRYSTALS. 

The  dark  rings  correspond  to  A  =  A,  2/,  3/1,  etc.,  and  since  (p.  100) 
the  indices  of  the  extraordinary  ray  are  alike  for  all  directions, 
equally  inclined  to  the  optic  axis,  n±  —  ;z,  must  be  constant  for  any 
one  value  of  i\  that  is,  these  rings  must  be  circles. 

The  cross  results  from  the  planes  of  vibration  of  the  different 
bundles  being  parallel  and  normal  to  different  principal  sections 
through  bundle  and  optic  axis.  For  any  one  position  of  the 
stage  the  vibration  planes  of  certain  bundles  will  be  in  the  diagonal 
positions  and  those  of  others  will  coincide  with  the  planes  of  the 
nicols ;  that  is,  the  light  circles  will  grade  from  greatest  brightness 
in  the  diagonal  positions  to  total  darkness  parallel  to  the  nicols  and 
as  the  stage  is  rotated  successive  rays  will  come  into  these  posi- 
tions, maintaining  the  same  effect. 

In  Sections  oblique  to  the  Optic  Axis  the  curves  must  be 
symmetrical  to  a  principal  section  through  the  plate  normal  and 
the  optic  axis.  If  the  optic  axis  shows  in  the  field  it  will  change 
its  position  with  rotation  of  the  stage,  but  the  arms  of  the  dark 
cross  will  always  remain  parallel  to  the  planes  of  the  nicols,  Fig. 
242.  One  arm  only  may  show. 


FIG.  242.  FIG.  243. 

hi  sections  parallel  to  the  optic  axis  the  curves  are  symmetrical  to 
the  principal  section  through  the  plate  normal  and  optic  axis  and  to 
a  plane  at  right  angles  to  the  optic  axis  and  are  conjugate  hyper- 
bolae, Fig.  243.  They  are  usually  only  visible  with  monochro- 
matic light. 
WITH  PARALLEL  NICOLS. 

The  change   from  crossed  to  parallel    nicols   reverses  all  the 
phenomena  of  interference,  p.   in.     When  J  =  /.,  2A,  3/,  etc.,  the 


OPTICAL  CHARACTERS.  117 

light  is  at  its  brightest  and  when  J  =  £/,  ^A,  |A,  etc.,  the  light  is 
extinguished.  That  is  with  the  quartz  wedge  the  thicknesses 
which  were  dark  bands  in  monochromatic  light  with  crossed  nicols 
will  be  light  and  the  light  bands  dark  and  with  white  light  the 
colors  produced  will  be  complementary  to  those  obtained  with 
crossed  nicols. 

With  convergent  light  all  bright  portions  of  the  field  are  ex- 
tinguished and  all  dark  portions  become  bright.  With  white 
light  the  complementary  colors  result. 

If  an  interference  color  is  passed  through  a  prism  the  spectrum 
will  show  dark  bands  in  the  positions  of  the  extinguished  colors 


Tests  with  Parallel  Light  and  Crossed  Nicols. 

The  polarizing  microscope,  p.  107,  is  nearly  always  used  in  de- 
termination, the  cross  hairs  being  made  parallel  to  the  previously 
determined  vibration  planes  of  the  nicols. 

DETERMINATION  OF  PLANES  OF  VIBRATION  OR  "  EXTINCTION." 

In  any  section  of  a  uniaxial  crystal  the  plane  of  vibration  of  the 
extraordinary  ray  passes  through  the  optic  axis  and  the  corre- 
sponding extinction  direction  is  the  projection  of  that  axis  on  the 
section  ;  the  plane  of  vibration  of  the  ordinary  ray  is  at  right  angles 
to  this. 

The  positions  of  maximum  darkness  or  extinction  directions 
are  always  either  parallel  or  symmetrical  to  cleavage  cracks  and 
crystal  outlines. 

A  color  contrast  is  more  easily  judged  and  may  be  obtained  either 
by  inserting  a  test  plate  of  quartz  or  gypsum  in  the  slot  always 
provided  between  the  nicols  or  by  use  of  a  special  eye-piece  such 
as  the  Bertrand.*  The  section  is  placed  to  cover  only  part  of  the 
field  and  for  the  extinction  positions  is  of  the  same  color  as  the 
rest  of  the  field,  but  for  any  other  position  there  is  a  difference  of 
tint. 

*  Four  quadrants  which  are  equally  thick  basal  sections  of  alternately  right  and 
left  handed  quartz;  the  lines  of  contact  take  the  place  of  the  cross  hairs.  Before  the 
introduction  of  the  crystal  section  the  four  quadrants  are  of  the  same  color. 

In  the  extinction  positions  the  crystal  section  is  colored  like  the  rest  of  the  field, 
but  the  slightest  divergence  from  this  raises  the  color  of  the  section  in  two  diagonally 
opposite  quadrants  and  lowers  the  color  in  the  other  two.  A  special  cap  nicol  must 
be  used  instead  of  ordinary  analyzer. 


1 1 8  CHARACTERS  OF  CR YSTALS. 

VIBRATION  DIRECTIONS  OF  FASTER  AND  SLOWER  RAYS. 

With  the  extinction  (vibration)  directions  in  diagonal  position,  a 
test  plate  of  some  mineral,  in  which  the  vibration  directions  have 
been  distinguished  and  marked,  is  inserted  between  the  nicols  (in 
a  slot  always  provided)  with  these  directions  also  diagonal.  If  the 
interference  color  is  thereby  made  higher,  the  vibration  direc- 
tions of  the  corresponding  rays  are  parallel ;  if  the  color  is  lowered, 
the  corresponding  directions  of  vibration  are  crossed. 

MICA  TEST  PLATE  OR  QUARTER  UNDULATION  MICA  PLATE. — A  thin  sheet  of  mica 
on  which  is  marked  c,  the  vibration  direction  of  the  slower  ray,  which  in  mica  is  the 
line  joining  the  optic  axes.  The  thickness  chosen  is  usually  that  corresponding  to  a 
blue  gray  interference  color  or  say  140^  which  is  £/l  for  a  medium  yellow. 

GYPSUM  TEST  PLATE  OR  GYPSUM  R,ED  OF  FIRST  ORDER. — A  thin  cleavage  of 
gypsum  on  which  is  usually  marked  a,  the  vibration  direction  of  the  faster  ray.  The 
thickness  chosen  corresponds  to  an  interference  color  of  red  of  first  order  or  say  $6ou/Li, 
which  is  essentially  /I  for  a  medium  yellow. 

QUARTZ  WEDGE. — A  thin  wedge  of  quartz,  cut  so  that  one  face  is  exactly  parallel 
to  the  optic  axis.  The  length  of  the  wedge  is  parallel  to  the  optic  axis,  and  as  quartz 
is  positive  this  direction  is  t,  the  vibration  direction  of  the  slower  ray. 

THE  V-FEDEROW  MICA  WEDGE. — Fifteen  quarter  undulation  mica  plates  superposed 
in  equivalent  position,  but  each  about  2  mm.  shorter  than  the  one  beneath  it. 

The  mica  plate  raises  or  lowers  the  value  of  J  by  one  quarter 
wave-length,  the  gypsum,  by  about  one  wave-length  and  the  two 
wedges  by  amounts  increasing  with  the  distance  inserted. 

When  sections  showing  only  low  colors  of  first  order  are  tested 
by  the  gypsum  plate,  there  is  practically  considered  the  effect  of 
the  plate  on  the  red  of  the  gypsum. 

DETERMINATION  OF  THE  RETARDATION*  A. 

The  v-Federow  mica  wedge  inserted  with  corresponding  vibra- 
tions directions  crossed  will  for  each  interposed  plate  reduce  A 

A 

by  140  fjL/j..     To  render  the  field  dark  will  require  -    -  =   ;/  plates. 

1 40 

Conversely  n.  140  =  /J,  in  which  n  is  determined  by  count. 

The  quartz  wedge  similarly  used  will  give  an  approximate  value 
by  counting  the  number  of  times  the  original  color  reappears,  if  ;/ 
times,  then  is  the  color  a  red,  blue,  green,  etc.,  of  n  -f  I  order,  for 
which  the  value  may  be  looked  up  in  a  chart. 

The  Babinet  Compensator  consists  of  two  equal  quartz  wedges 

*  Difference  in  Retardation. 


OPTICAL  CHARACTERS.  119 

A  and  B,  Fig.  244,  so  cut  that  the  optic  axis  c  of  say  A  is  parallel 

to  the  length  y  and  that  of  B  is 
parallel  to  the  breadth  z.  Any 
normally  incident  ray  /  will  be 
divided  in  B  into  a  faster  (ordi- 
i  ;,  nary)  ray  vibrating  parallel  toj/ 

and   slower   (extraordinary)    vi- 
brating parallel  to  z.    But  reach- 
ing A  the  faster  ray  becomes  the  slower,  and  vice  versa. 

For  the  central  position  these  will  exactly  balance,  and  with 
•either  monochromatic  or  white  light  there  will  be  here  a  dark  band 
with  which  a  cross  hair  is  made  to  coincide,  and  on  each  side,  with 
monochromatic  light,  there  will  be  at  equal  distances  from  this  other 
parallel  dark  bands  corresponding  to  A  =  A,  2A,  3/,  etc. 

Let  A  be  made  moveable  by  a  micrometer  screw  and  B  be  fixed 
and  denote  the  movement  necessary  to  bring  the  second  dark  band 
into  coincidence  with  the  cross-hair  by  d,  then  d  corresponds  to  A  of 
the  light  used,  and  a  movement  of  nd  corresponds  to  a  difference 
of  retardation  of  «A.  , 

With  the  compensator  in  the  zero  position  and  diagonal  to  the 
planes  of  the  nicols,  introduce  a  mineral  section  also  in  diagonal 
position  and  determine  the  motion  D  necessary  to  bring  back  the 
central  band  under  the  cross-hair,  this  change  being  due  to  the 

mineral,  measures  the  value  of  A  in  the  section,  that  is,  A  =  — -   in 

0 

wave-lengths,  or  —A  in  millionths  of  millimeter  //  /*. 
d 

If  on  the  scale  used,  8  is  unity,  then  D  =  A  in  wave-lengths. 

DETERMINATION  OF  THE  STRENGTH  OF  THE  DOUBLE  REFRACTION. — If 
the   thickness    of    the    section    is   known   (n^  —  ;/)   results    from 
A  =  t(n^  —  n). 
DETERMINATION  OF  THICKNESS  OF  SECTION. 

From  formula  A  =  /(;/t  —  n),  t  may  be  calculated  if  the  value  of 
n^  —  n  is  known  either  for  the  crystal  or  for  fragments  of  other 
known  minerals  ground  with  the  section. 

Moderately  thick  sections  (0.5  mm.  and  upwards)  may  be  meas- 
ured by  fastening  on  a  thin  cover  glass  larger  than  the  section  with 
thin  balsam,  cleaning  away  the  balsam  at  the  edges  by  alcohol 
and  focusing  successively  on  dust  upon  the  lower  surface  of  the 
cover  and  the  upper  surface  of  the  glass.  The  difference  in  ele- 
vation is  measured  by  the  micrometer  focussing  screw  and  is  t. 


120 


CHARACTERS  OF  CRYSTALS. 


Measurements  with  a  micrometer  eye-piece,  the  section  being  set 
on  edge  are  sometimes  possible. 
APPROXIMATE  DETERMINATION  OF  PRINCIPAL  INDICES. 

The  method  of  the  Due  de  Chaulnes*  as  improved  by  Sorbyf 
depends  upon  the  fact  that  the  focal  distance  of  a  microscope  is 
altered  when  a  plane-parallel  plate  is  inserted  between  the  objec- 
tive and  the  focus.  .  Sorby  focussed  upon  fine  lines  ruled  on  glass 
and  placed  some  distance  below  the  objective.  Using  the  lower  nicol 
only  the  indices  corresponding  to  rays  of  definite  vibration  direction 
were  determined  either  by  measuring  the  displacement  due  to  the 
unmounted  section  or  successively  those  due  to  the  glass  alone 
and  glass  plus  section.  The  thickness  of  the  section  and  the  dis- 
placement due  to  one  revolution  of  the  micrometer  focussing  screw 
must  be  known.  Then  denoting  the  thickness  by  t  and  the  dis- 
placement by  d 


11,  —  n   -  i 

t—  d 
in  which  if  the  outer  medium  is  air  n1  =  I. 

Becke  determines  the  relative  indices  of  two  substances  in  con- 
tact in  a  section  by  focussing  upon  the  dark  boundary  line  and 
raising  the  telescope  tube,  upon  which  the  dark  boundary  appears 
to  move  towards  the  substance  with  the  higher  index  of  refraction. 

TESTS  WITH  CONVERGENT  LIGHT. 
DETERMINATION  OF  CHARACTER  J  OF  RAY  SURFACE. 

The  MICA  TEST  PLATE  inserted  diagonally  above  a  section  normal 


FIG.  245.  FIG.  246. 

to  the  optic  axis  will  destroy  the  black  cross  and  break  the  rings 

*Mem.  de  I  Acad.  Paris,  1767-68.    zz^  \Mineral  Mag.  I.,  193,  II.  I. 

JWhen  the  direction  of  the  optic  axis  is  known  the  character  may  be  determined 
in  parallel  light  by  the  extinction  direction  which  is  the  projection  of  the  optic  axis. 
If  this  correspond  to  the  slower  ray,  p.  1 18,  the  crystal  is  positive,  if  to  the  faster  ray  the 
crystal  is  negative. 


OPTICAL  CHARACTERS.  121 

into  four  quadrants,  the  relative  effects  in  positive  and  negative 
crystals  being  shown  in  Figs.  245,  246.  The  corresponding  signs 
_j_  and  —  are  suggested  by  the  relative  positions  of  the  dark  flecks 
and  the  arrow  showing  the  direction  t  of  the  test  plate. 

After  insertion  of  the  test  plate  a  vertical  plane  through  t  will 
contain  the  vibration  directions  of  the  slower  ray  in  the  mica,  of 
the  extraordinary  rays  in  the  quadrants  passed  through  (first  and 
third)  and  of  the  ordinary  rays  in  the  other  quadrants  (second  and 
fourth). 

In  positive  crystals  the  ordinary  ray  being  the  faster  there  will  be 
an  increase  of  1/4  /  in  the  first  and  third  quadrants  and  a  decrease 
of  1/4  A  in  the  second  and  fourth. 

At  the  centre  J  will  now  be  1/4  A  and  no  longer  dark.  In  the 
first  and  third  quadrants  the  distances  between  the  rings  will  be 
decreased  about  one-fourth  by  the  increase  of  J  by  1/4  L  In  the 
second  and  fourth  quadrants  by  the  lessening  of  A  the  spaces  be- 
tween the  rings  will  be  increased  and  near  the  centre  the  portion 
formerly  bright  with  J=i/4  /  will  become  J=o;  that  is,  two  new 
dark  flecks  will  be  developed  as  in  Fig.  245. 

In  negative  crystals  the  extraordinary  ray  being  the  faster  the 
phenomena  are  exactly  reversed.  The  centre  is  light,  the  rings 
are  narrowed  in  the  second  and  fourth  quadrants  and  widened  in 
the  first  and  third  and  in  these  two  dark  flecks  are  developed  near 
the  centre  as  in  Fig.  246. 

THE  GYPSUM  RED  OF  FIRST  ORDER  inserted  with  a  parallel  to  the 
arrow  is  particularly  advantageous  for  very  thin  or  feebly  re- 
fracting sections  in  which  the  rings  are  almost  out  of  the  field. 
The  centre  will  be  red.  In  positive  crystals  the  first  and  third 
quadrants  will  lower  the  red  to  say  yellow  and  in  the  second  and 
fourth  near  the  centre  will  raise  the  red  to  blue.  In  negative 
crystals  the  reverse  will  take  place ;  that  is,  the  "  blue  quadrants  " 
correspond  in  position  to  the  black  flecks.  This  determination 
must  be  made  in  white  light. 

BY  SUPERPOSITION  OF  A  BASAL  SECTION  OF  A  MINERAL  OF  KNOWN 
SIGN.  If  the  two  sections  are  alike  in  character  this  will  simply 
act  like  a  thickening  of  the  plate.  If  unlike  they  will  partially 
neutralize  each  other.  This  test  is  little  used. 


CHAPTER  IX. 


THE  OPTICALLY   UNIAXIAL    CRYSTALS  (Continued), 


IV.     OPTICALLY  UNIAXIAL    CRYSTALS  IN  WHICH  THE  OPTIC 
AXIS  IS  A  DIRECTION  OF  CIRCULAR  POLARIZATION. 

In  certain  hexagonal  and  tetragonal  crystals  monochromatic 
plane  polarized  light  transmitted  in  the  direction  of  the  crystal 
axis  c  (optic  axis)  is  not  extinguished  by  crossed  nicols,  but  is  ex- 
tinguished after  a  definite  rotation  of  the  analyzer. 

PLANE,  CIRCULAR  AND  ELLIPTICAL  POLARIZATION. 

If  light  with  vibrations  in  one  plane  is  decomposed  into  light 
with  vibrations  in  two  planes  at  right  angles,  any  ether  particle  in 
the  path  of  two  interfering  rays,  p.  106,  will  be  acted  upon  by  two 
vibrations  which  will  be  at  right  angles,  but,  generally  speaking, 
these  will  be  neither  of  the  same  phase  nor  of  the  same  intensity. 
The  effects  are  analogous  to  those  produced  upon  a  swinging 
pendulum  by  a  second  impulse  at  right  angles  to  the  first. 

i°.  In  general  the  resultant  motion  is  elliptical  and  the  light  is  ellip- 
tically  polarized.  For  instance,  the  light  emerging  from  a  doubly 
refracting  plate  in  a  polariscope  is  in  general  elliptically  polarized, 

2°.  If  the  phase  difference  is  a  half 
vibration  (J/2  A),  or  an  entire  vibration, 
the  resulting  motion  is  in  a  straight 
line  and  the  light  is  plane  polarized. 
For  example,  Fig.  247,  if  AU  is  the 
original  direction,  OD  and  OC,  the 
components  at  right  angles  y2  A  apart, 
the  resultant  motion  will  be  El  equal 
AU,  but  not  in  the  same  direction. 

3°.  If  the  vibrations  are  of  equal 
amplitude,  but  possess  a  phase  differ- 
ence of  y±  A,  that  is,  if  one  vibration  is 
at  the  center  of  its  swing  when  the  other  reverses  its  direction,  the 
resultant  motion  is  circular  and  the  light  is  circularly  polarized. 
For  example,  yellow  rays  emerge  from  a  quarter  undulation  mica 
plate  in  the  polariscope  circularly  polarized. 


FIG.  247. 


OPTICAL  CHARACTERS. 


123 


In  the  Fresnel  rhomb,  Fig.  248,  the  index  of  refraction  of  the 
glass  and  the  angles  of  the  rhomb  are  so  related  that  a  plane 
polarized  ray  S,  normally  incident,  experiences  a  total  reflection 
at  E  and  another  at  U  at  such  angles  that  at  each  reflection 
there  is  developed  a  phase  difference  of  y&  A  |between  the  com- 
ponent vibrating  in  the  plane  of  reflection  and  that  vibrating  in  a 
plane  at  right  angles  thereto.  That  is,  the  vibration  of  any 
particle  in  the  emerging  ray  T  is  the  result  of  two  component  vi- 
brations at  right  angles  with  a  phase 
difference  of  ^  A,  but  not  in  general  of 
equal  intensity.  The  ray  is  therefore 
elliptically  polarized  and  if  examined  by 
a  nicol  will  show  variations  in  inten- 
sity, but  never  complete  extinction. 

If  the  plane  of  vibration  of  the  in- 
cident light  is  at  45°  to  the  plane  of 
reflection  of  the  rhomb,  the  component 
vibrations  will  be  equal  in  intensity, 
the  emerging  light  will  be  circularly 
polarized,  and  the  analyzing  nicol  will 
show  no  variation  in  intensity. 

If  two  rhombs  are  used,  the  four  successive  total  reflections  de- 
velope  a  phase  difference  of  y2  A,  that  is,  the  impulses  are  in  the 
'Same  straight  line,  the  emerging  light  is  plane  polarized  and  is 
extinguished  whenever  its  plane  of  vibration  coincides  with  that 
of  the  long  diagonal  of  the  nicol. 

ROTATION  OF  THE  PLANE  OF  POLARIZATION. 

Two^cfrcular  vibrations  in  opposite  directions  in  the  same  plane 
are  equivalent  to  a  straight  lined  vibration  of  double  amplitude 
and  vice  versa.  A  particle  at  A,  Fig.  249,  simultaneously  sub- 
jected to  two  motions  tending  to  carry  it  in  any  time  /,  at  a  uni- 
form angular  velocity  through  the  equal  arcs  AR  and  AL  would 
evidently  in  this  time  travel  along  the  path  AU  =  2AS,  for  the 
components  SL  and  SR  at  right  angles  to  AS  are  equal  and  oppo- 
site and  of  no  effect.  For  the  arcs  ART  and  ALT,  the  distance 
travelled  would  be  AN '  =  2AT.  Conversely,  the  vibration  AN  is 
equivalent  to  the  two  circular  motions  ART  and  ALT. 

If  a  plane  polarized  ray  be  decomposed  by  any  substance  into 
two  rays  circularly  polarized  in  opposite  directions,  these  rays,  in 

*  Due  to  difference  in  velocity  and  length  of  path. 


124 


CHARACTERS  OF  CRYSTALS. 


traversing  a  plate  of  given  thickness,  will  develope  a  phase  differ- 
ence, *  and  on  emergence  will  combine  to  a  plane  polarized  ray. 

The  planes  of  vibration  (or  of  polarization,  see  p.  100)  of  the  in- 
cident and  emerging  rays  will  not  be  parallel. 

Let  AN,  Fig.  250,  represent  the  vibration  of  the  incident  ray, 
R  and  L  the  advance  points  of  the  emerging  rays  at  the  moment 


of  emergence,  then  will  DF  be  the  vibration  of  the  emerging  ray, 
D  being  midway  between  R  and  L,  for  only  for  this  direction  will 
the  components  SL  and  SR  neutralize  each  other. 

The  rotation  for  a  thickness  of  I  mm.  is  usually  denoted  by  a  and 
for  any  other  thickness  /,  by  /  «.  The  value  of  «  in  terms  of  the  wave 
length  and  the  indices  of  refraction  of  the  two  circularly  polarized 


ray  s* 


s      a=— 


THE  OPTIC  Axis  A  DIRECTION  OF  DOUBLE  REFRACTION. 

In  this  division  of  uniaxial  crystals,  any  incident  plane-polarized 
ray  transmitted  in  the  direction  of  the  optic  axis,  is  converted  into  two 
diverging  rays  circularly  polarized  in  the  opposite  sense  (direction). 

From  the  formula  above  given,  the  divergence  n"—nr  may  be 
calculated,  or,  even  when  minute  as  in  quartz,  it  may  be  experi- 
mentally shown  by  the  method  of  Fresnel  with  three  prisms  of 
right  and  left  quartz  cut  as  in  Fig.  251. 

In  A  P  D  the  right-handed  ray  is  the  faster,  reaching  PA  Cit 
is  the  slower,  hence  is  bent  towards  the  normal  while  the  left-handed 

*Th.  Liebisch.     Grundriss  dcr  Fhys.  Kryst.  1896,  p.  124. 


OPTICAL  CHARACTERS. 


125 


ray  is  bent  away  from  the  normal.  Reaching  ACE  the  right  is 
again  the  faster  and  is  bent  from  the  normal  while  the  left  is  bent 
towards  it,  and  finally  on  emerging  both  are  bent  from  the  normal : 
all  changes  acting  to  increase  the  divergence  until  with  PA  C= 
152°  a  divergence  of  4°  is  reached  which  can  be  measured  by  a 
goniometer. 

C  P 


It  can  be  shown  by  a  nicol  that  these  emerging  rays  are  circu- 
larly polarized  and  in  the  opposite  sense. 

RAY  SURFACE. 

The  ray  surface  is  closely  that  of  the  preceding  division,  but  with 
the  ellipsoid  and  sphere  not  quite  in  contact  at  the  optic  axis,  one 
of  them  a  little  flattened,  the  other  a  little  drawn  out.* 

It  is  probable  that  there  is  a  grading  from  plane  polarized  light 
transmitted  normal  to  the  optic  axis,  through  elliptically  polarized 
light,  the  eccentricity  of  the  ellipse  decreasing  with  the  angle  be- 
tween the  transmission  direction  and  the  optic  axis,  to  circularly 
polarized  light  in  the  direction  of  the  optic  axis. 

The  optic  axis  remains  an  axis  of  isotropy,  every  diameter  perpen- 
dicular to  it  is  an  axis  of  binary  symmetry,  but  there  are  no  planes  of 
symmetry  because  the  direction  of  rotation  in  corresponding  shells 
at  the  extremities  of  the  optic  axis  are  reversed. t 

The  division  is  therefore  limited  to  crystals  in  which  the  crystal 
faces  are  not  symmetrical  either  to  any  plane  or  to  the  centre.  A 
further  condition  seems  to  be  that  the  right  and  left  forms  shall  be 
"  enantiomorphs  "  (the  one  can  not  by  rotation  about  any  axis  be 
brought  into  coincidence  with  the  other).  For  example,  Group  17 
has  no  planes  of  symmetry,  but  the  forms  are  not  enantiomorphs 
and  circular  polarization  does  not  occur. 

*  Lang  with  quartz  shows  the  velocity  of  the  ordinary  rays  near  the  axis  is  not  quite 
constant  and  that  the  extraordinary  does  not  follow  exactly  the  Huyghens  law  of  com- 
mon uniaxial  crystals. 

•f  The  sense  of  rotation  is  not  reversed  on  inverting  the  plate. 


126  CHARACTERS  OF  CRYSTALS. 

The  classes  in  which  circular  polarization  is  possible  are : 
TETRAGONAL  SYSTEM.  Class  9,  to  which  no  crystals  have  yet 
been  referred.  Class  10,  circular  polarization  is  not  proved  in  any 
crystal  of  this  class.  Class  12,  examples:  strychnine  sulphate,, 
guanadin  carbonate,  diacetylphenol-thalein.  HEXAGONAL  :  (R.HOM- 
BOHEDRAL  DIVISION).  Class  1 6,  example:  sodium  periodate,  Na 
IO4.  3H2O.  Class  1 8,  examples:  quartz,  cinnabar,  matico  cam- 
phor C12H20O,  tartrates  of  rubidium  and  caesium,  subsulphates  of 
lead,  potassium,  calcium  and  strontium ;  (HEXAGONAL  DIVISION). 
Class  23,  example:  potassium-lithium  sulphate,  K  Li  SO4.  Class 
24,  circular  polarization  is  not  proved  in  any  crystal  of  this  class. 

The  direction  of  rotation  corresponds  to  the  right  and  left  of 
the  enantiomorphs.* .  With  one  or  two  exceptions,^. £•., sulphate  of 
strychnine,  circular  polarization  is  destroyed  by  dissolving,  fusing 
or  vaporizing  the  crystal.  Certain  substances,  however,  showing 
in  crystalline  condition  no  power  to  rotate  the  plane  of  polariza- 
tion show  it  in  solution  or  in  vapor,  and  are  called  optically  active. 
These  substances  when  in  crystals  are  enantiomorphic  right  or  left 
as  the  solution  is  right-handed  or  left-handed.  If  acids  are  optic- 
ally active  their  salts  usually  are  so.f 

WITH  PARALLEL  MONOCHROMATIC  LIGHT  AND  CROSSED  NICOLS. 

In  sections  normal  to  the  Optic  Axis  the  field  will  be  illuminated,, 
though  only  feebly  in  thin  sections,  and  will  only  become  dark  when 
the  analyzer  has  been  turned  an  amount  equal  to  the  rotation 


That  is,  an  amount  increasing  with  the  thickness  and  double  re- 
fraction in  the  direction  of  the  optic  axis  and  decreasing  with  the 
wave-length.  Approximately 

A      B 


*  Herschel  in  Trans.  Cambridge  Soc.  I,  43,  showed  that  circular  polarization  in 
quartz  was  apparently  produced  by  the  same  cause  which  determined  the  position  of 
the  so-called  plagihedral  faces,  for  a  plate  of  quartz  cut  perpendicular  to  the  vertical 
axis  of  a  crystal  in  which  the  left  plagihedral  faces  occur  will  turn  the  plane  of  polar- 
ization of  the  incident  ray  to  the  left  and  a  similar  plate  from  a  crystal  showing  the 
right  plagihedral  faces  will  turn  this  plane  to  the  right. 

f  See  Pasteur's  lectures  before  Societe  chimique  de  Paris  1860,  published  as  No.  28, 
Ostwald's  Klassiker  der  Exacten  Wissenschaften.  Ueber  die  Asymmetric.  Leipzig,. 
1891. 


OPTICAL  CHARACTERS. 


127 


In  all  other  sections  there  is  extinction  every  90°  and  brightest 
illumination  in  the  diagonal  positions.  Extinction  also  takes  place 

for       =  i,  2,  3,  etc.,  as  described  p.  in. 

WITH  PARALLEL  WHITE  LIGHT  AND  CROSSED  NICOLS. 

In  sections  normal  to  the  optic  axis,  since  the  rotation  increases  as 
the  wave-length  decreases,  the  different  colors  will  be  dispersed 
and  emerge  vibrating  in  different  planes.  As  the  analyzer  is 
turned  its  plane  of  vibration  will  be  successively  at  right  angles  to  the 
planes  of  vibration  of  the  different  colors.  That  is,  at  any  time 
only  one  color  can  be  shut  out  completely  and  the  rest  in  varying 
degree.  The  color  with  vibrations  most  nearly  parallel  to  the 
vibration  planes  of  the  analyzer  will  be  least  shut  out  and  will  de- 
termine the  predominating  tint  in  the  color  seen. 

In  all  other  sections  there  will  be  extinction  every  90°  and  great- 
est brightness  in  diagonal  positions  and  the  color  tint  due  to  inter- 
ference as  described  p.  113. 

WITH  CONVERGENT  LIGHT  AND  CROSSED  NICOLS. 

In  sections  normal  to  the  optic  axis,  with  monochromatic  light,  thin 
sections  will  show  the  dark  cross 
the  arms  of  which  will  always  be 
parallel  to  the  vibration  planes  of 
the  nicols  and  somewhat  indis- 
tinct near  the  centre.  In  thicker 
sections  the  bars  will  not  reach 
the  centre,  Fig.  252,  and  no  dark 
rings  will  form  there,  the  first  ring 
being  some  distance  out,  no  mat- 
ter how  thick  the  section  may  be, 
beyond  this  there  will  appear  the 
alternate  dark  and  light  rings 
with  radii  nearly*  as  in  the  last 
division. 

In  general  the  inner  circle  is  bright  except  when  the  thickness  for 
the  light  used  gives  la.  =  180°  x  «,  in  which  n  =  I,  2,  3,  etc.,  when 
it  is  black.  With  white  light  the  central  circle  has  the  color  tint 

*  Exact  measurements  show  that  the  radii  of  the  circles  near  the  centre  do  not  obey 
the  law  of  common  uniaxial  crystals.  J.  C.  McConnell,  Phil.  Trans.  Roy.  Soc,  1886, 
177,  299.  1887. 


FIG.   252. 


128 


CHARACTERS  OF  CRYSTALS. 


which  is  obtained  over  the  entire  field  when  parallel  light  is  used. 
On  turning  the  analyzer  each  ring  is  decomposed  into  four  arcs, 
which  widen  when  the  direction  is  that  proper  to  the  section  and 
contract  when  it  is  the  opposite. 

In  sections  oblique  to  the  axis  essentially  as  described  p.  1 16. 

In  sections  parallel  to  the  optic  axis  essentially  the  hyperbolic 
curves  of  Fig.  243. 

Determination  of  Optical  Characters. 

Most  of  the  determinations  are  exactly  as  described  under  the 
preceding  division.     References  may  therefore  be  made  as  follows; 
Direct  determination  of  principal  indices,  p.   103  ;   Extinction  direc- 
tions, p.  117;  Faster  and  slower  ray,  p.  118;  Retardation,  p.  118 
Strength  of  double  refraction,  p.  119. 

DETERMINATION  OF  DIRECTION  OF  ROTATION  OF  PLANE  OF  VIBRA- 
TION. 
In  a  section  normal  to  the  optic  axis  and  with  crossed  nicols : 

(a)  By  the  widening  of  the  arcs  when  the  analyzer  is  turned  in 
the  direction  of  rotation. 

(b)  A  quarter  undulation  mica  plate  inserted  diagonally  above 


FIG.  253. 


FIG.  254. 


the  plate  converts  the  rings  and  cross  into  two  intervvound  spirals, 
which  start  near  the  centre  and  wind  in  the  direction  of  rotation,* 
Fig.  253,  Right;  Fig.  254,  Left.  If  the  mica  is  placed  below  the 
plate  the  spirals  are  reversed. 

*  A  plate  of  left  and  another  of  right-handed  substance  give  a  fourfold  spiral  in 
which  the  direction  of  winding  conforms  to  the  lower  plate. 


OPTICAL  CHARACTERS. 


129 


(c)  The  direction  in  which  it  is  necessary  to  turn  the  analyzer  to 
change  the  tint  of  passage  (see  below)  into  red  is  the  direction 
of  rotation. 

DETERMINATION  OF  ANGLE  OF  ROTATION  OF  PLANE  OF  VIBRATION. 

In  a  section  normal  to  the  optic  axis  and  with  crossed  nicols : 

(a)  With  monochromatic  light  the  rotation  of  the  analyzer  neces- 
sary to  produce  darkness  is  la. 

(b)  With  white  light  and  Tint  of  Passage  or  Sensitive  Tint.     When 
the  analyzer  is  at  right  angles  to  the  brilliantly  coloring  yellow 
green,  for  which  A  =  550 /AM  in  air,  the  remaining  colors  unite  to  an 

easily  recognized  grayish  violet, 
called  the  tint  of  passage  or  sensi- 
tive tint,  because  the  slightest  fur- 
ther rotation  in  the  one  direction 
uer"'ed  gives  red  and  in  the  other  blue. 
g«  Starting  with  the  polarizer  and  an- 
yell  alyzer  parallel,  the  angle  turned  by 
the  analyzer  to  produce  the  tint  is 
la  for  A=55o/Jt/Jt  (middle  yellow). 
Fig.  255  shows  the  rotations  pro- 
duced by  a  quartz  plate  of  3.75  mm. 
thickness.  With  parallel  nicols  the 
vibration  plane  of  yellow  is  then  at 

right  angles  to  PS,  the  vibration  plane  of  the  analyzer,  that  is  yellow 
is  extinguished  and  the  tint  of  passage  results.  With  crossed  nicols 
the  tint  of  passage  results  when  the  vibration  plane  for  yellow  is 
PS,  that  is,  for  a  rotation  of  180°  or  a  thickness  of  7.50  mm. 

(c)  With  white  light  and  spectrum. 

The  light  polarized  in  P,  Fig.  256,  passes  through  the  substance 
vS,  the  collimator  C,  the  analyzer  A  and  through  a  slit  and  is  de- 
composed by  the  prism  R  and  viewed  by  the  telescope  T.  As  the 
analyzer  is  turned 'a  dark  line  corresponding  to  the  extinguished 
ray  will  move  along  the  spectrum  and  the  angles  corresponding  to 
the  rotations  (from  crossed  position)  which  bring  this  line  into  co- 
incidence with  the  Frauenhofer  lines  will  correspond  to  la.  for  the 
corresponding  values  of  A. 

If  a  quartz  plate  is  placed  at  0,  the  light  passes  through  the 
quartz,  analyzer,  slit  and  decomposing  prism  and  yields  a  spectrum 


FJG.  255. 


130  CHARACTERS  OF  CRYSTALS. 

with  a  dark  line.     The  analyzer  is  turned  until  the  dark  line  coin- 


FIG.  256. 

•tides  with  a  Frauenhbfer  line;  then  the  substance  is  introduced  and 
the  rotation  necessary  to  bring  the  two  lines  again  into  coinci- 
dence is  the  value  of  /«  for  that  color. 

If  a  biquartz,*  Fig.  257,  of  7.50  mm.  thick- 
ness be  used,  with  crossed  nicols,  or  one  of  3.75 
mm.  with  parallel  nicols,  a  black  line  will  appear 
for  yellow  (A  =  550  P.;J.\  which  for  slight  rotations 
of  the  analyzer  will  move  into  the  blue  or  the  red. 
With  the  introduction  of  the  substance  at  5  the 
black  line  is  disturbed  and  the  rotation  necessary 
F  G-  257-          to  re-establish  it  is  the  value  of  la  for  yellow. 

ABSORPTION  AND  PLEOCHROISM. 

In  uniaxial  crystals  with  monochromatic  light  the  two  rays  cor- 
responding to  any  direction  of  transmission  are  in  general  absorbed 
at  a  different  rate.  The  faster  ray  may  be  either  more  absorbed  or 
less  absorbed  than  the  slower,  the  absorption  of  the  ordinary  ray 
being  independent  of  the  direction  of  transmission,  while  that  of 
the  extraordinary  varies  with  the  inclination  to  the  optic  axis,  but 
is  constant  for  the  same  angle  and  differs  most  from  that  of  the 
ordinary  for  transmission  normal  to  the  optic  axis. 

With  white  light  the  color  seen  in  any  direction  will  be  due  to 
a  combination  of  the  ordinary  and  extraordinary  rays,  and  this 
combination  will  give  different  tints  in  different  directions. 

Usually  the  two  rays  are  viewed  separately  by  means  of  a  polar- 
izing microscope  with  the  analyzer  out.  The  color  varies  as  the 

*Two  semicircular  basal  plates  of  right  and  left  quartz,  of  equal  thickness 
(Biot's  3.75  mm.,  Soleil's  7.50  mm.)  and  with  their  twinning  plane  either  parallel  or  at 
right  angles  to  the  vibration  plane  of  the  polarizer. 


OPTICAL  CHARACTERS.  131 

stage  is  turned  and  the  maximum  color  differences  are  obtained 
when  the  extinction  directions  coincide  with  the  vibration  plane 
of  the  polarizer  (shorter  diagonal). 

In  sections  normal  to  the  optic  axis  the  color  is  constant  for  all 
positions  of  the  stage. 

In  sections  parallel  to  the  optic  axis.  If  the  extinction  direction 
parallel  to  the  optic  axis  is  placed  first  parallel  then  at  right 
angles  to  the  short  diagonal  there  will  be  seen  first  the  extraordi- 
nary, second  the  ordinary. 

In  all  other  sections  the  extraordinary  will  be  found  to  approach 
the  constant  tint  of  the  ordinary  as  the  sections  become  more 
nearly  perpendicular  to  the  optic  axis. 

The  colors  of  ordinary  and  extraordinary  may  be  contrasted 
side  by  side  by  means  of  a  "  dichroscope"  consisting,  Fig.  258,  of 


f 


' 


e        b  d 

FIG.  258. 

a  rhomb  of  calcite  in  which  ab  and  cd  are  the  short  diagonals  of 
opposite  faces.  To  these  faces  glass  wedges  aeb,  dfc  are  cemented 
and  the  whole  encased.  The  section  is  placed  at  P  and  the  light 
from  the  substance  passes  through  a  rectangular  orifice,  a  double 
image  of  which  is  seen  by  the  eye  at  E. 

When  the  planes  of  vibration  of  the  section  and  the  calcite  coin- 
cide the  two  rays  originated  in  the  section  will  have  the  maximum 
color  difference  of  the  section. 

If  the  plate  is  turned  around  the  axis  of  the  dichroscope  each 
ray  from  the  crystal  is  decomposed  in  the  calcite  and  contributes 
a  portion  of  its  intensity  to  each  of  the  two  images  and  at  45°  each 
will  contribute  half  of  its  brightness  to  each  image  ;  that  is,  the 
the  two  images  will  then  be  alike  in  color  and  brightness. 

Basal  sections  of  dark  tourmaline  or  of  magnesium  platinocyan- 
ide  show  phenomena  analogous  to  the  absorption  tufts  described 
under  biaxial  absorption. 


CHAPTER  X. 


THE  OPTICALLY  BIAXIAL  CRYSTALS. 


In  any  orthorhombic,  monoclinic  or  triclinic  crystal  the  ray  sur- 
face for  light  of  any  definite  wave-length  is  symmetrical  to  three 
planes  at  right  angles  to  each  other  which  intersect  in  the  vibration 
directions  of  the  fastest  and  slowest  rays,  the  third  intersection 
being  the  vibration  direction  of  a  ray  of  some  intermediate  velocity. 

The  planes  of  symmetry  are  called  Optical  principal  sections  and 
their  intersections  are  called  Principal  vibration  directions.*  No 
diameter  is  an  axis  of  isotropy,  that  is  a  direction  about  which  the 
crystals  are  optically  equivalent,  but  in  two  directions  there  is  sin- 
gle refraction  and  these  directions  from  analogy  with  the  uniaxial 
are  called  the  "  Optic  axes  ''for  light  of  that  wave-length. 

As  in  the  uniaxial  crystals  the  relative  values  of  the  principal  in- 
dices of  refraction,  that  is,  the  indices  of  the  rays  with  the  principal 
vibration  directions,  change  with  the  wave-length.  But  as  a 
consequence  the  directions  of  the  optic  axes  change  also, 
the  amount  of  change  or  so  called  "  dispersion  "  varying  from  a 
few  minutes  to  over  forty  degrees.  That  is,  the  optic  axes  in  this 
group  of  crystals  are  not  fixed  in  position  as  in  the  uniaxial. 

With  decreasing  geometric  symmetry  not  even  the  optical 
principal  sections  or  principal  vibration  directions  remain  fixed  in 
direction,  but  alter  with  the  change  of  light.  Upon  this  so-called 
"  Dispersion  of  the  Bisectrices  "  are  based  optical  distinctions  be- 
tween the  systems. 

In  Orthorhombic  Crystals  the  geometric  axes  are  always  princi- 
pal vibration  directions  and  the  axial  planes  are  optical  principal 
sections  for  all  colors. 

In  Monoclinic  Crystals  the  ortho  axis  b  is  always  a  principal  vi- 
bration direction  and  the  clinopinacoid  oio  is  an  optical  principal 
section  for  all  colors,  the  other  two  principal  vibration  directions 
lie  in  oio,  but  vary  in  position  with  the  wave-length. 

*Often  called  axes  of  elasticity. 


OPTICAL  CHARACTERS. 


133 


In  Triclinic  Crystals  there  are  no  principal  vibration  directions  or 
principal  sections  which  are  constant  for  all  colors. 

THE  OPTICAL  INDICATRIX. 

The  inductive  method  by  which  the  present  conception  of  a  biax- 
ial ray  surface  has  been  reached  may  be  briefly  outlined  as  follows  : 

It  was  shown  by  Huyghens  that  the  ray  surface  of  calcite  was  a 
double  surface  composed  of  a  sphere  and  an  ellipsoid  of  revolu- 
tion, and  the  ray  surfaces  of  other  tetragonal  and  hexagonal  sub- 
stances were  found  to  be  similar  and  symmetrical  always  to  the 
crystallographic  planes  of  symmetry.  It  was  later  shown  that 
from  an  ellipsoid  of  revolution  all  the  relations  between  direction 
of  transmission,  velocity  and  vibration  direction  could  be  deduced. 

By  analogy  it  was  conceived  that  for  any  less  symmetrical  crys- 
tal some  correspondingly  less  symmetrical  figure  or  Optical  Indi- 
catrix  existed  from  which  these  relations  and  the  shape  of  the  ray 
surface  could  be  geometrically  determined.  Experiment  has 
proved  this  to  be  true,  and  that  for  orthorhombic,  monoclinic  and 
triclinic  crystals  the  optical  indicatrix  is  an  ellipsoid  the  three  rec- 
tangular axes  of  which  are  in  length  proportionate  to  the  so-called 
principal  indices  of  refraction. 

Let  a  <  /?  <  Y  be  these  principal  indices  of  refraction  and  a,  b  and 
c  the  directions  of  vibration  of  the  corresponding  rays,  that  is,  of 
the  rays  with  greatest,  intermediate  and 
least  velocity  of  transmission.  Upon  the 
directions  a,  b,  c  lay  ofif  <9a  =  «,  Ob  =  ft, 
and  <9c  =  y  and  upon  these  as  axes  con- 
struct an  ellipsoid  that  is  the  Optical  In- 
dicatrix, Fig.  259.  Then,  by  analogy 
with  the  uniaxial,  p.  102,  for  any  diam- 
eter considered  as  a  direction  of  trans- 
mission there  are  two  points  of  the  sur- 
face the  normals  from  which  are  also 
normal  to  that  diameter.  These  normals 
are  at  once  the  directions  of  vibration  and 
the  reciprocals  of  the  velocities  of  the  rays 
transmitted  in  the  direction  of  the  diameter-. 

RAY  SURFACE. 

The  shape  of  the  ray  surface  may  be  judged  from  the  shape  of 
the  optical  principal  sections  ab,  cb  and  ac. 


134 


CHARACTERS  OF  CRYSTALS. 


Optical  Principal  Section  ab.  For  the  direction  aOt  Fig.  259,  the  two 

normals  are  b(9  and  c(9  corresponding  to  velocities-  and  - ;  for  the 

A        r 

direction  bO  the  normals  are  aO 
and  c<9,  corresponding  to  velocities 

—  and  -,  for  any  other  direction 
a  r 

one  normal  is  c<9,  the  other  a  line 
in  ab,  found  as  described  on  p.  102 
and  varying  in  length  between  aO 
and  W,  according  to  the  direction 
of  the  ray,  like  the  radius  vectors 
of  an  ellipse.  Hence  for  any  di- 
rection of  transmission  one  ray  vi- 
brates in  some  direction  in  ab,  but 
is  transmitted  with  a  velocity  vary- 
ing like  the  radius  vectors  of  an 

ellipse  between  -  in  the  direction  b  to  -  in  the  direction  a,  Fig. 

260,  and  corresponds  to  an  extraordinary  ray. 

The  other  ray  for  any  direction  vibrates  in  the  direction  of  c,  nor- 
mal to  ab  and  is  transmitted  with  constant  velocity-.  It  corresponds, 
therefore,  to  an  ordinary  ray. 
Hence  for  ab  the  sections  of  the 
two  shells  are  as  in  Fig.  260. 

Optical  Principal  Section,  cb. 
In  precisely  the  same  manner  it 
is  shown  that  for  any  direction 
of  transmission  in  the  plane  cb 
one  ray  is  extraordinary  in  that 
its  vibration  direction  is  always 
b  and  its  velocity  varies  between 

— ,  in  the  direction  b,  to  I  in  the 

r 

direction  c  ;  whereas  the  other 

ray  is  ordinary  with  a  constant 

vibration  direction  a  normal  to 


FIG.  260. 

I 
a 


FIG.  261. 


cb  and  a  constant  transmission  velocity  — .    That  is  for  cb  the  sec- 

a 

tions  of  the  two  shells  are  as  in  Fig.  261. 


OPTICAL  CHARACTERS. 


135 


Optical  Principal  Section  ac  or  Plane  of  tJie  Optic  Axes.  In  this 
case  for  any  direction  of  transmission  one  ray  is  extraordinary 
with  its  vibration  direction  in  ac  and  velocity  varied  between 

-  in  the  direction  a  and  -  in  the  direction  c,  while  for  the  ordinary 
ray  the  constant  vibration  direction  is  &  and  the  constant  velocity 
-*  That  is,  in  this  section  of  the  two  shells,  there  must  be  four 

symmetrically  placed  points  of  in- 
tersection £,  as  shown  in  Fig.  262. 
Each  principal  section  then  con- 
sists of  an  ellipse  and  a  circle.  The 
entire  surface  may  be  generated  as 
follows. 

Revolve  the  section  ac  about  one 
of  its  axes  a,  the  curves  of  the  sec- 
tion touching  always  those  in  the 
section  ab  and  altering  according 
to  a  definite  law  until  after  90° 
rotation  they  coincide  with  the 
curves  cf  the  section  . 
OPTIC  AXES.* 

In  Section  ac  of  the  indicatrix, 
Fig-  259»  there  must  be  a  radius 

vector  OL  equal  /?,  for  these  vary  between  a.  and  Y  •  Similarly 
OLJ  (not  shown)  symmetrically  opposite  OL  will  equal  'ft.  The 
sections  of  the  indicatrix  through  OL  Ok  or  OL'  Ob  must  be  circles. 
In  section  ac  of  the  ray  surface,  Fig.  262,  near  but  not  at  the 
points  E,  common  tangent  planes  can  be  drawn  to  each  shell. 

The  directions  normal  to  these  circular  sections  in  the  indicatrix 
and  to  the  common  tangent  planes  in  the  ray  surface  are  the  so- 
called  optic  axes  A  A  of  Fig.  262. 

A  circular  section  of  the  indicatrix  is  the  common  ray  front  of  all 
the  diameters  determined,  p.  133,  by  the  normals  to  the  indicatrix 
from  each  point  of  the  circle.  Of  these  normals  only  one,  Ob,  is  m 
the  circle,  hence  only  one  diameter  is  normal  to  the  circle  (the  optic 

*  The  directions  EE  are  variously  called  Ray  Axes,  Secondary  Axes,  and  Bira- 
dials  and  are  simply  directions  of  equal  ray  velocity,  the  ray,  however,  corresponding 
to  an  infinite  number  of  planes  tangent  at  £,  the  normals  to  which  diverge  and  form 
a  cone  (outer  conical  refraction). 


FIG.  262. 


136 


CHARACTERS  OF  CRYSTALS. 


axis),  while  the  others  starting  from  0  diverge  and  form  a  cone. 
Furthermore  since  the  normals  are  all  in  different  directions  the 
rays,  transmitted  in  the  directions  of  the  diameters;  have  all  di- 
rections of  vibration.  The  same  facts  are  evident  from  Fig.  262 ; 
the  common  tangent  plane  caps  the  conical  depression  at  E. 
From  each  point  of  the  circle  of  contact  the  lines  to  0  represent 
rays  with  a  common  ray  front  and  of  which  the  front  normal  is 
the  optic  axis.  That  is,  each  optic  axis  is  the  front  normal  for 
a  diverging  cone  of  rays  (inner  conical  refraction)  which  on  emer- 
gence are  parallel  to  this  front  normal  but  each  of  which  has  a  dif- 
ferent direction  of  vibration. 

The  velocity  in  the  direction  of  an  optic  axis  is  evidently  -^-  —  — 

C/b      p 

POSITIVE  AND  NEGATIVE  RAY  SURFACES. 

In  the  plane  of  the  optic  axes  a  and  c  bisect  the  angles  between 
these  axes.  If  the  acute  bisectrix  is  c,  (Bxa  =  c),  that  is,  if  bisects 
the  acute  angle  between  the  optic  axes  the  ray  surface  is  said  to  be 
positive. 

If  the  acute  bisectrix  is  a,  (Bxa  =  a),  that  is,  if  a  is  the  acute  bisec- 
trix and  c  the  obtuse,  the  ray  surface  is  said  to  be  negative.  This 
conforms  strictly  to  the  usage  in  the  uniaxial  which  is  a  special 
case  of  biaxial  with  the  angle  between  the  optic  axes  zero,  that  is 
with  one  optic  axis  in  the  direction  c. 

REFRACTION  IN  BIAXIAL  CRYSTALS. 

In  general,  following  the  Huyghens  construction,  p.  86,  the  new 

ray  fronts  through  the  point  E,  Fig. 
208,  are  tangent  to  the  shells  in  points 
which  do  not  lie  in  the  plane  of  inci- 
dence, that  is,  both  rays  are  bent  out 
of  the  plane  or  are  extraordinary. 
With  normal  incidence  this  is  still 
true.  Let  SURT,  Fig.  263,  be  the 
section  of  the  indicatrix  normal  to 
the  ray.  The  planes  ROf  and  TOf 
through  the  axes  of  the  ellipse  and  the 
front  normal  contain  the  two  rays 

Oe  and  Ot  at  right  angles  to  the  surface  normals  RE  and   UN. 

These  normals  are  the  vibration  direction  of  the  rays  and  are  not 


FIG.  263. 


OPTICAL  CHARACTERS. 


137 


at  right  angles  to  each  other,  though  on  emergence  both  rays  retake 

the  direction  Of  with  vibrations  at  right  angles  and  a  phase  difference. 
Because  SfJRTis  normal  to  the  incident  ray  it  is  parallel  to  the 

plate,  hence  the  extinction  directions  in  the  plate  are  OR  and 

OU  the  projections  of  the  vibration  directions. 

If  the  plane  of  incidence  be  a  principal  sec- 
tion then  the  points  of  tangency  of  the  new 
ray  fronts  will  be  in  the  plane.  One  will  for 
all  directions  show  a  constant  refraction  in- 
dex and  may  be  called  the  ordinary. 

WITH  PARALLEL  MONOCHROMATIC  LIGHT  AND 
CROSSED  NICOLS. 

In  sections  normal  to  an  optic  axis  the  field 
will   maintain   a  uniform  brightness   during 
rotation  of  the  stage,  because  p.  136,  in  the 
emerging  cylinder  of  rays  developed  by  in- 
ner conical  refraction  each  ray  vibrates  in  a  different  plane. 

In  all  other  sections  the  field  is  dark  every  90°  and  is  illuminated 
for  all  other  positions  and  most  brilliantly  in  the  diagonal  posi- 
tions. The  field  is  also  dark  throughout  an  entire  revolution  for 


-  =  1,2,  3,  etc.,  p. 


in,  and  brightest  for  —  = 


f,  ^»  etc->  m 


conforming  to  similar  uniaxial  sections. 

Relation  between  Extinction  Directions  and  Optic  Axes.  As  just 
explained,  the  extinction  directions  are  the  diameters  JTand  HH, 
Fig.  264,  of  that  central  section  of  the  indicatrix  which  is  parallel 
to  the  plate.  The  two  circular  sections  must  intersect  any  central 
section  in  equal  diameters  tt  which  from  the  nature  of  an  ellipse 
must  be  symmetrical  to  the  axis  of  the  ellipse,  that  is,  to  the  ex- 
tinction directions. 

If  planes  be  passed  through  the  plate  normal  and  each  optic  axis 
their  traces  ee  will  be  at  90°  to  those  of  the  circular  sections,  hence 
also  symmetrical  to  the  extinction  directions.  That  is,  the  extinc- 
tion directions  must*  bisect  the  angle  between  the  traces  of  the  two 
planes,  each  through  the  plate  normal  and  an  optic  axis. 

WITH  PARALLEL  WHITE  LIGHT  AND  CROSSED  NICOLS. 

In  sections  normal  to  an  optic  axis  for  light  of  one  color  there  is 

*Upon  this  principal  and  after  a  device  by  Prof.  Groth,  B5hm  and  Wiedemann,  of 
Munich,  construct  a  model  for  graphic  determination  of  extinction  directions. 


138 


CHARACTERS  OF  CRYSTALS. 


constant  illumination  throughout  rotation  because  of  internal  con- 
ical refraction  p.  1 36 ;  the  optic  axes  for  other  colors  are,  however, 
more  or  less  oblique  to  the  section  and  double  refraction  takes 
place,  and  as  the  resulting  vibration  directions  are  somewhat  dif- 
ferently oriented  for  each  color,  color  tints  result  somewhat  as  in 
circular  polarization. 

In  all  other  sections.  If  there  is  no  marked  dispersion  of  the 
principal  vibration  directions  there  will  be  approximately  perfect 
extinction  every  90°  and  interference  colors  as  in  the  uniaxial,  but 
with  marked  dispersion  the  vibration  directions  in  any  plate  di- 
verge for  the  different  colors  and  the  plate  can  never  be  perfectly 
dark  and  moreover  in  any  position  the  interference  colors  due  to 
phase  difference  are  modified  by  color  tints  due  to  the  partial  or 
complete  extinction  of  certain  colors. 

WITH  CONVERGENT  MONOCHROMATIC  LIGHT  AND  CROSSED  NICOLS. 

In  sections  normal  to  a  or  c  there  will  be  darkness  whenever  the 
vibration  planes  of  the  emerging  rays  coincide  with  those  of  the 

nicols  and  whenever  --  is  a  whole   number;  see   p.  in.     The 

A 

points  of  emergency  of  the  optic  axis  will  therefore  be  dark,  since 
—  =  o,  and  the  points  corresponding  to  — '  =  I  will  together  form 

A  A 

a  ring  around  each  axis  and  similarly  for  values  of  2,  3,  4,  etc., 
until  the  pair  corresponding  most  nearly  to  the  value  —  for  the  cen- 
tre of  the  field*  unite  at  or 
near  the  centre  to  a  cross  loop 
or  figure  eight  around  both 
axes  and  subsequent  rings 
form  lemniscates  around  this 


as  in  Fig.  265.     If—  for  the 


FIG.  265. 


centre  is  less  than  unity  even 
the  first  ring  must  surround  both  axes,  giving  them  a  figure  more 

*For  the  rays  transmitted  normally  at  the  center  of  the  field  —  =—  (n, — «).  If 

A  A 

the  direction  of  transmission  is  c  them  #x  —  n=  y—,3  and  if  direction  is  a  then  nl  —  n 
—  ft  —  a.  In  either  case  —  is  known  and  determines  the  number  of  rings  between 
the  axes  and  the  centre. 


OPTICAL  CHARACTERS. 


139 


FIG.  266. 


similar  to  that  of  a  uniaxial  crystal.  As  all  parts  of  the  lemniscates 
are  independent  of  positions  of  planes  of  vibration,  there  will  be 
no  change  during  rotation  of  the  stage. 

When  the  stage  is  rotated  so  that  principal  sections  are  parallel 
to  the  vibration  planes  of  the  nicols  all  the  rays  transmitted  in 
those  sections  (therefore  vibrating 
in  or  at  right  angles  to  those  sec- 
tions) will  be  extinguished  and 
there  will  be  a  sharp  dark  band 
joining  the  poles  and  another  some- 
what thicker  lighter  band  at  right 
angles  to  the  first  and  midway  be- 
tween the  poles.  These  are  often 
called  brushes.  For  any  other  po- 
sition of  the  stage  the  vibrations  of 
some  other  rays  are  parallel  to  those 
of  the  nicols,  and  these  are  always 
so  distributed  that  the  resultant 
dark  spots  form  the  branches  of  an 
hyperbola  through  each  of  the  axes, 
effect  is  as  in  Fig.  266. 

As  the  stage  is  rotated  the  straight  bars  appear  to  dissolve  into 
the  hyperbola,  the  branches  of  which  appear  to  rotate  in  the  oppo- 
site direction  to  the  stage,  the  convex  side  always  toward  the  other 

axis.     If—  is  nowhere  equal  to  I  there  will  be  no  dark  rings,  but 

A 

the  hyperbola  will  appear  as  before.  Because  the  two  principal 
sections  are  planes  of  symmetry  the  curves  and  brushes  are  at 
all  times  symmetrical  to  the  traces  of  the  sections. 

In  sections  normal  to  S  the  dark  curves  due  to  —  =  I,  2,  3*  etc., 

are  hyperbolae  similar  to  those  shown,  Fig.  243.  No  brushes  are 
visible. 

///  sections  normal  to  an  optic  axis  there  will  be  the  same  forma- 

j 
tion   of  dark   rays  corresponding  to  whole   numbers    lor  —.the 

A 

curves,  however,  will  be  more  nearly  circles,  Fig.  267.  There  will 
be  a  straight  black  bar  bisecting  the  curves  whenever  the  trace  of 
the  plane  of  the  optic  axes  coincides  with  the  vibration  direction 
of  either  nicol,  and  for  all  other  angles  of  rotation  there  will  be 


At  the  diagonal  position  the 


140 


CHARACTERS  OF  CRYSTALS. 


one  arm  of  an  hyperbola  through  the  axis,  Fig.  268,  the  convex 
side  towards  the  other  axis.  This  arm  will  rotate  in  the  opposite 
direction  to  the  stage. 


FIG  267. 


FIG.  268. 


WITH  CONVERGENT  WHITE  LIGHT  AND  CROSSED  NICOLS. 

In  sections  normal  to  the  acute  bisectrix  Because  of  the  dispersion 
of  the  optic  axes  described,  p.  130,  the  centres  of  the  ring  systems 
are  not  the  same  for  the  different  colors.  If  there  is  also  a  dis- 
persion of  the  bisectrices  the  section  can  only  be  exactly  normal 
to  the  bisectrix  for  one  color.  These  facts,  added  to  the  general 
one  that  for  different  colors  the  greater  the  wave-length  the  greater 
the  distance  apart  of  the  rings,  render  it  possible  in  white  light  to 
determine  the  crystalline  system  by  the  distribution  of  the  colors, 
thus  obtaining  Divisions  V,  VI  and  VII. 

V.    ORTHORHOMBIC    CRYSTALS. 

The  color  distribution  must  be  symmetrical  to  the  line  joining 
the  optic  axes  and  to  the  line  through  the  centre  at  right  angles 
thereto,  because  these  are  the  traces  of  the  principal  sections  com- 
mon to  the  ray  surfaces  of  all  the  colors.  Assuming  that  the  same 
principal  section  is  the  axial  plane  for  all  colors  there  may  be  two 
cases  :  i°.  The  axes  for  the  longer  wave-lengths  may  be  most  dis- 
persed, that  is,  the  angle  between  the  optic  axes  for  red  greater 
than  that  for  violet,  or  p  >  o.  2°.  The  axes  for  the  shorter  wave- 
lengths may  be  most  dispersed,  that  is,  the  angle  between  the  optic 
axes  for  red  less  than  that  for  violet,  or  p  <  o. 

Figure  269  illustrates  the  second  class.  In  the  interference 
figure  for  the  diagonal  position  the  hyperbolae  will  pass  through 
the  axial  points  for  each  color  and  in  each  that  color  will  be  ex- 
tinguished, giving,  if  the  dispersion  is  great,  a  series  of  colored 
bands  ranging  from  red  (white  minus  violet  and  part  of  blue) 


OPTICAL  CHARACTERS. 


141 


through  the  violet  axes ;  to  blue  or  violet  (white  minus  red  and 
part  of  yellow)  through  the  red  axes.      Generally  the  intermediate 
colors  overlap,  producing  a  dark  band,  except  on  the  outer  fringes. 
That  is,  the  colors*  fringing 
the  hyperbola   are   in  in- 
verse position  to  the  axial 
points. 

Similarly,  if  the  full  line 
circles  represent  the  first 
extinction  ring  for  violet 
and  the  dotted  those  for 
red,  then  as  we  pass  from 
the  centre  red  is  first  ex- 
tinguished and  the  violet 
tint  shown. 

For  />  <  L>  the  red  is  fur- 
ther from  the  centre  than 
the  blue.  For  f>^>o  the 
red  is  nearer  the  centre 
than  the  blue.  Or  in  gen- 
eral, the  color  with  the 
larger  angle  is  nearer  the 
centre  of  the  field. 

If  the  axial  plane  is  not 
the  same  for  all  colors,  that 
is,  if  b  for  certain  colors 
becomes  a  or  c  for  others, 
then  the  interference  fig- 
ures will  be  turned  90° 
with  respect  to  each  other 
and,  if  superposed  in  white 
light,  will  give  a  complex 
figure  as  in  brookite. 


FIG.  269. 


VI.     MONOCLINIC  CRYSTALS. 

There  must  also  be  considered  the  dispersion  of  the* bisectrices. 
Three  general  cases  have  been  distinguished,  in  all  of  which  oio  is 
a  principal  section  and  the  ortho  axis  b  a  principal  vibration  direc- 
tion. 


*  The  dotted  curves  should  be  colored  blue  ;  the  full  line  curves  red,  in  Figs.  269- 
272. 


142 


CHARACTERS  OF  CRYSTALS. 


V     R 


(a)  Inclined  Dispersion.     The  optic  axes  for  all  the  colors  lie  in 
the  plane  oio  and  the  ortho  axis  b  is  6,  the  intermediate  principal 
vibration  direction,  but  a  and  c  vary  in  position  in  the  plane  oio. 
Evidently  the  color  distribution  will  be  symmetrical  to  oio. 

Let  Fig.  270  represent 
the  plane  oioin  a  section 
cut  normal  to  the  acute 
bisectrix  for  yellow  light. 
The  acute  bisectrices  for 
red  and  violet  will  be  ob- 
lique to  this  section  as 
represented.  Assume 
P  >  a  and  construct  the 
interference  hyperbolae 
and  first  ring  for  the  di- 
agonal positions.  Four 
things  are  at  once  no- 
ticed : 

I  °.  The  color-distribu- 
tion and  shape  are  sym- 
metrical to  the  trace  of 
oio  and  to  this  line*  only. 

2°.  The  combination 
color  rings  around  the 
axes  due  to  overlapping 
of  rings  for  the  different 
colors  are  very  different, 
the  one  relatively  small, 
circular  and  intense,  the 
other  larger,  oval  and  re- 
latively dull.  The  se- 
^JG  2  Q  quence  of  colors  may  be 

the  same  or  reversed. 

3°.  The  smaller  ring  is  further  from  the  centre  of  the  field. 
4°.  One  hyperbola  band  is  much  broader  than  the  other.  ^ 

(b)  Horizontal  Dispersion.     If  the  ortho  axis  b  is  the  acute  bisec- 
trix (either  a  or  c  as  the  crystal  is  -J-  or  — )  then  the  planes  of  the 
optic  axes  must  for  all  colors  pass  through  it  and  be  normal  to  the 
plane  oio.     In  a  section  cut  normal  to  the  acute  bisectrix  for  yel- 
low light  let  Fand  R,  Fig.  27 1,  represent  the  axial  planes  for  violet 

*  Line  connecting  the  axial  points. 


OPTICAL  CHARACTERS. 


143 


and  red,  rr  the  optic  axes  for  red,  vv  those  for  violet.     The  plane 
oio  (not  shown)  is  normal  to  all  the  axial  planes. 

In  the  interference  figure  it  is  evident  that  the  color-distribution 
and  shape  of  the  lemniscates  are  symmetrical  to  the  trace*  of  oio 
but  not  to  any  other  line.  Hence,  for  normal  position  the  hori- 


FIG.  271. 

zontal  dark  bar  will  be  centrally  black  from  overlapping  colors,  but 
on  the  edges  will  be  red  and  violet  or  blue. 

(c)  Crossed  Dispersion.  If  the  ortho  axis  b  is  the  acute  bisectrix 
(either  c  or  a  as  the  crystal  is  positive  or  negative)  then  will  the 
section  normal  to  the  acute  bisectrix  for  one  color  be  normal  for 
all  and  will  itself  be  parallel  to  oio.  The  axial  planes  for  all  colors 
will  pass  through  b,  and  as  b  corresponds  to  a  or  c  it  is  an  axis  of 
binary  symmetry,  hence  the  color-distribution  and  shape  of  the 
lemiscates  must  be  such  that  any  straight  line  through  the  centre 
of  the  field  passes  in  each  direction  through  similar  points.  That 
is,  diagonally  opposite  parts  of  the  field  are  similar.  In  Fig.  272 
Fand  R  are  the  axial  planes  for  violet  and  red. 

It  will  be  noticed  that  the  chief  reliance  throughout  is  upon  the 


*  Normal  at  centre  to  line  joining  the  axes. 


144 


CHARACTERS  OF  CRYSTALS. 


colors  that  fringe  the  horizontal  brush  or  the  hyperbola,  and  that 
this  is  in  every  case  the  other  end  of  the  spectrum  to  the  color  of 
the  light  producing  it. 


FIG.  272. 
VII.    TRICLINIC  CRYSTALS. 

With  triclinic  crystals  there  is  no  symmetry  at  all  in  the  arrange- 
ment of  the  colors  in  the  interference  figure  from  white  light. 
Two  or  more  varieties  of  dispersion  may  occur  at  once. 

Plates  normal  to  the  obtuse  bisectrix  show  similar  phenomena, 
though  frequently  it  is  impossible  to  obtain  the  interference  figures 
within  the  field.  The  color-distribution  is  naturally  changed  as 
the  dispersion  of  axes  is  in  the  opposite  direction. 


CHAPTER   XI. 


DETERMINATION  OF  THE    OPTICAL    CHARACTERS 
OF  BIAXIAL   CRYSTALS. 


Some  of  the  determinations  are  exactly  as  described  under  the 
uniaxial.  References  may  therefore  be  made  of  follows  :  Faster  and 
sl&iverray,  p.  146  ;  Retardation,  p.  f7f§  ;  Strength  of  double  refraction, 
p.  frzjT*-;  Thickness  of  section  ,  p.  147.  V  I  % 


DETERMINATION  OF  PLANES  OF  VIBRATION  OR  "  EXTINCTION." 

In  biaxial  crystals  complete  extinction  in  general  is  only  ob- 
tained with  monochromatic  light,  p.  138.  The  methods  of  deter- 
mination are  as  described,  p.  1  37. 

In  sections  normal  to  one  of  a,  B,  or  c  the  extinction  directions 
are  the  other  two  principal  vibration  directions.  In  sections  par- 
allel to  a,  6,  or  c  one  extinction  direction  must  be  a  principal  vibra- 
tion direction.  In  sections  normal  to  an  optic  axis  no  extinction 
directions  are  obtained.  The  general  relation  between  extinction  di- 
rections and  optic  axes  is  described,  p.  1  37.  This  test  is  often  suffi- 
cient to  determine  the  crystalline  system,  especially  when  an  entire 
crystal  can  be  examined  and  the  extinction  directions  determined 
in  all  faces  and  zones. 

In  ortJwrhombic  crystals  extinction  will  always  be  parallel  or  sym- 
metrical to  crystallographic  edges,  cleavage  cracks,  etc.  In  pina- 
coidal  faces  the  vibration  direction  will  be  parallel  to  the  crystal 
axes. 

In  monoclinic  crystals  there  will  be  parallel  or  symmetrical  ex- 
tinction in  the  zone  [ooi  100]  and  in  this  zone  the  vibration  direc- 
tions in  the  base  and  orthopinacoid  will  be  parallel  to  the  axes.  In 
all  other  zones  the  extinction  will  be  oblique. 

In  triclinic  crystals  all  extinctions  will  be  oblique. 

THE   ORIENTATION  AND  DISTINCTION  OF  THE   PRINCIPAL  VIBRA- 

TION DIRECTIONS. 

If  the  positions  of  the  optic  axes  are  known  then  all  three  prin- 
cipal vibration  directions  a,  b  and  c  are  also  known,  for  a  and  c  are 


146  CHARACTERS  OF  CRYSTALS. 

the  bisectrices  of  the  angles  between  the  optic  axes  and  6  is  normal 
to  their  plane.  The  indices  a,  ft  and  f  may  then  be  determined,  as 
described  later,  and  the  indicatrix  and  ray  surface  therewith  con- 
structed. 

In  orthorhombic  crystals  the  pinacoids  are  normal  to  a,  b  and  c. 
That  normal  to  b  will  show  with  convergent  monochromatic  Itght 
hyperbolae*  similar  to  Fig.  243  and  those  normal  to  a  and  c  will 
show  lemniscates,  Fig.  270. 

In  monoclinic  crystals  the  clinopinacoid  is  necessarily  either  nor- 
mal to  b  and  shows  the  hyperbolae,  Fig.  243,  with  convergent 
monochromatic  light,  or  is  normal  to  a  or  c  and  shows  lemniscates. 
The  extinction  directions  therein  are  the  other  two  principal  vi- 
bration directions,  and  the  faster  and  slower  ray  may  be  distin- 
guished as  described,  p.  146.  Plates  may  be  ground  normal  to 
these  directions. 

In  triclinic  crystals  by  trial  a  face  is  found  which  with  convergent 
monochromatic  light  shows,  even  eccentric,  the  hyperbolae  or 
lemniscates  or  axial  image ;  from  this  the  direction  is  judged  in 
which  a  new  face  may  be  ground  exactlyf  normal  to  a  principal 
vibration  direction.  The  extinction  directions  in  the  new  face  will 
be  the  other  principal  vibration  directions  and  the  faster  and  slower 
rays  corresponding  may  be  distinguished  as  on  p.  146.  Sections 
may  be  ground  normal  to  them.  If  there  is  dispersion  of  bisec- 
trices accurate  work  will  require  special  sections  for  each  color,  if 
the  dispersion  is  not  large,  sections  for  a  middle  color,  usually  soda 
yellow,  are  used. 

DIRECT  MEASUREMENT  OF  THE  PRINCIPAL  INDICES  OF  RERFACTION. 

(a]  Prisms  with  a  refracting  edge  B,  Fig.  212,  parallel  to  a 
principal  vibration  direction  will  give  for  minimum  deviation,  p. 
8&,  a  direction  of  transmission  RS  so  lying  in  an  optical  principal 
section  that  the  ray  vibrating  parallelj  to  the  edge  B  will  yield  a 
principal  index.  In  general  the  other  obtainable  index  is  not  a 
principal  index. 

By  Fig.  259  it  is  seen  that  if  B  is  parallel  to  c  the  transmission 

*  By  means  of  one  of  the  attachments  to  the  polarizing  microscope  described  later, 
sections  approximately  normal  to  a,  b  or  c  may  be  tipped  so  that  the  lemniscates  or 
hyperbolae  are  obtained. 

fit  is  important  to  leave  faces  of  known  forms  so  that  later  the  ground  faces  may  be 
goniometrically  oriented. 

JThe  ray  which  penetrates  a  nicol  when  the  short  diagonal  is  parallel  to  B. 


OPTICAL  CHARACTERS.  147 

is  in  ab  and  the  ray  vibrating  parallel  to  B  yields  ^.  Similarly  if 
B  is  parallel  to  b,  ft  results;  and  if  B  is  parallel  to  a,  a.  results. 

If  the  bisector  BD,  Fig.  212,  is  also  the  trace  of  an  optical  prin- 
cipal section  then  the  direction  of  transmission  RS  for  minimum 
deviation  is  also  a  principal  vibration  direction  and  both  rays  yield 
principal  indices. 

(6)  Prisms  in  which  BD,  Fig.  212,  the  bisector  of  the  refracting 
angle  B  is  a  principal  vibration  direction  also  give  for  minimum 
deviation  a  direction  of  transmission  RS  lying  in  an  optical  prin- 
cipal section.  Therefore  the  index  of  the  ray  vibrating  parallel  to 
BD,  that  is  at  right  angles  to  the  edge  B,  is  a  principal  index. 

By  Fig.  259  it  is  seen  that  if  BD  is  parallel  to  c  the  transmission 
RS  being  normal  to  BD  lies  in  the  plane  normal  to  c,  that  is  in  the 
optical  principal  section  ba  and  the  ray  vibrating  parallel  to  BD 
or  at  right  angles  to  B  will  yield  Y.  Similarly  if  BD  is  parallel  to 
B  the  ray  vibrating  at  right  angles  to  B  will  yield  ft  and  if  BD  is 
parallel  to  a  the  ray  vibrating  at  right  angles  to  B  will  yield  «. 

If  a,  b,  or  c,  bisect  the  exterior  angle  B,  Fig.  212,  then  is  the 
bisector  BD  normal  to  this  principal  vibration  direction  and  RS 
parallel  thereto,  that  is  for  minimum  deviation  the  direction  of 
transmission  is  a  principal  vibration  direction  and  both  rays  yield 
principal  indices.  The  formulae  under  (/;),  p.  103,  hold  for  (a)  and 

0). 

,  (c]  If  one  face  AB,  Fig.  228,  of  a  prism  is  an  optical  principal 
section  and  the  refracting  edge  A  is  a  principal  vibration  direction, 
rays  normally  incident  at  AB  will  be  transmitted  in  the  direction 
RS  parallel  to  another  principal  vibration  direction  and  each  ray 
will  yield  a  principal  index.  If  AB  is  ab  and  A  is  either  a  or  b, 
Fig.  259,  then  RS  is  c  and  a  and  ft  result.  The  formulae  under  (c), 
p.  103,  hold  good. 

(d)  In  any  crystal  face  or  section  parallel  to  a  principal  vibration 
direction,  by  the  methods  and  formulae  of  pp.  90-95,  if  this  direction 
is  made  the  direction  of  transmission  the  two  limit  lines,  as  in  p. 
104,  correspond  to  principal  indices ;  for  instance  if  the  transmission 
direction  is  a  the  two  limit-lines  correspond  to  ft  and  f.  If  the 
plate  is  turned  90°  in  its  own  plane  the  direction  of  transmission 
will  be  normal  to  a  principal  vibration  direction  (to  a  in  case  given) 
and  the  limit-line  of  the  ray  the  vibration  direction  of  which  is  at 
right  angles  to  this  direction  of  transmission  will  correspond  to 
the  third  index. 


1 48  CHAR  A  CTERS  OF  CR  YS  TALS. 

If  the  plate  is  parallel  to  a  principal  section*  then  either  prin- 
cipal vibration  direction  used  as  a  direction  of  transmission  will 
give  two  of  the  principal  indices. 

DETERMINATION  OF  THE  ANGLE  BETWEEN  THE  OPTIC  AXES. 

This  is  usually  accomplished  in  a  plane  parallel  plate  cut  nor- 
mal to  the  acute  bisectrix.  In  orthorhombic  crystals  the  same 
plate  will  be  normal  for  all  colors,  but  in  the  other  systems  this  is 
not  so,  but  if  the  plate  be  cut  normal  for  a  middle  color,  say  yel- 
low, the  results  for  all  colors  will  be  approximatelyf  accurate. 
The  apparent  angle,;);  denoted  by  2£,  is  larger  than  the  true  angle, 
denoted  by  2  V,  and,  since  light  transmitted  in  the  direction  of  the 
optic  axis  has  the  middle  index  /?,  p.  136,  then 


If  2E  exceed  180°  the  plate  is  usually  surrounded  by  oil  or  some 
other  dense  medium  in  a  vessel  with  plane  parallel  sides.  Denot- 
ing the  angle  then  obtained  by  2H  and  the  index  of  the  surround- 
ing medium  by  n 

n  sin  H 

- 


By  measuring  a  second  plate  normal  to  the  obtuse  bisectrix 
there  results 

T7.      nsvsiE' 

sin  V  —  — 


but  as  2F-f2F=i8o°  then  V+  V  -=90°  and  sin  V  —  cos 
whence 

sin  V      n  sin  Eft  sin  E 

°r  tan  V  ''= 


n  sn 


*lf  the  plate  be  parallel  to  the  optic  axes  the  two  boundaries  approach  as  the  plate 
is  turned  and  meet  at  the  optic  axis.  In  fact  the  shape  of  both  sections  of  the  Fres- 
nel's  surface  may  be  plotted  and  even  the  optic  axial  angle  measured.  Liebisch, 
Grundriss  der  Phys.  Kryst.,  373. 

f  The  rays  can  only  emerge  at  the  true  angle  when  the  optic  axes  are  normal  to  the 
surface,  as  in  a  sphere  or  in  a  cylinder  the  axis  of  which  is  perpendicular  to  the  axial 
plane;  or  when  the  outer  medium  has  the  same  index  of  refraction  as  the  ray  trans- 
mitted in  the  direction  of  the  optic  axis.  The  former  is  rarely  practicable,  the  latter 
will  be  described  later. 

\  Plates  normal  to  obtuse  bisectrix  yield  V  =  900  —  V. 


OPTICAL  CHARACTERS. 


149 


This  measurement  involves  actual  rotation  of  the  plate  about  b  as 
an  axis  between  the  lenses  of  a  polariscope,  the  plane  of  rotation 
being  at  45°  to  the  vibration  planes  of  the  crossed  nicols  and  the 
arms  of  the  hyperbola  being  brought  in  succession  into  tangency 
with  one  cross  hair.  The  line  joining  the  optic  axes  in  the  images 
must  coincide  with  the  other  cross  hair. 


The  Groth  Universal  Apparatus.  The  optical  portion  of  the 
polariscope  shown,  Fig.  235,  is  placed  in  horizontal  position  in 
another  stand,  Fig.  273.  The  nicols  are  crossed  at  45°  to  the 
horizon  and  just  enough  space  left  between  the  objectives  to 
permit  free  rotation  of  the  crystal  around  a  vertical  axis.  Above 
is  a  horizontal  graduated  circle  A"  with  concentric  central  rings  R 
and  F  which  in  turn  support  the  vertical  crystal  carrier.  The  lat- 
ter has  motions  of  adjustment  for  the  crystal  plate  which  in  the 
latest  type  of  instrument  are  exactly  those  of  the  Fuess  Goniometer, 
p.  67,  but  more  frequently  are  simple  sliding  motions  of  a  plate  at 
f  and  spherical  segment  ("  mushroom  joint ")  at  H.  In  this  simpler 
type  the  mineral  is  centered  by  drawing  the  front  tube  back  until 
the  crystal  is  in  focus,  then  centering  as  in  the  goniometer,  then 


ISO  CHARACTERS  OF  CRYSTALS. 

the  images  are  centered  by  the  mushroom  joint  until  both  stay  on  a 
horizontal  line. 

The  closer  the  lenses  the  larger  the  image :  that  is,  a  small  crys- 
tal should  be  used,  and  as  the  best  results  are  obtained  with  rather 
narrow  interference  rings  the  crystals  should  be  of  such  a  thick- 
ness (selected  or  obtained  by  grinding)  that  the  lemniscates  are 
distinct.  In  measuring  in  a  denser  medium*  the  glass  vessel  J/is 
used  and  the  lenses  touch  its  walls. 

It  is  usually  a  sufficiently  accurate  selection  of  a  plane  normal 
to  a  bisectrix  if  the  center  of  the  lemniscate  coincide  with  the  cen- 
ter of  the  field.  Groth  givesf  methods  for  more  accurate  deter- 
mination of  the  plane. 

The  No.  2  Fuess  Goniometer,  Fig.  193,  maybe  used  for  the  meas- 
urement of  axial  angles  by  substituting  nicols  for  the  signal  tube 
in  the  collimator  and  for  the  eye  piece  in  one  of  the  telescopes  ; 
both  nicols  are  adjustable.  A  tube  containing  a  condensing  lens 
is  fitted  over  the  objective  end  of  the  collimator  like  a  cap  and 
the  telescope  is  converted  into  a  weak  microscope  by  the  extra 
lens  used  to  bring  a  crystal  into  •  focus.  The  crystal  section 
is  carefully  mounted  as  previously  described.  The  telescope 
is  set  opposite  the  collimator,  the  nicols  being  crossed  45°  to 
the  plane  of  rotation  of  the  goniometer  and  rotation  of  the 
carefully  centered  crystals  brings  the  two  branches  of  the  hyper- 
bola successively  into  contact  with  the  vertical  cross  hair. 

By  a  combination  of  the  axial  angle  apparatus  with  a  spectro- 
scopej  the  angle  for  all  portions  of  spectra  can  be  determined. 

A  Polarizing  Microscope  may  be  used,  the  interference  figure  be- 
ing seen : 

(a]  By  placing  an  extra  lens§  (Klein's  lens)  above  the  eye  piece, 
thus  making  visible  a  little  image  which  forms  there,  or 

(&}  By  placing  below  the  eye  piece  but  above  the  focal  plane  of 
the  objective  a  weak  magnifying  lens  (Bertrand  lens). 

Two  methods  of  measuring  are  possible  : 

(a)  Measurement  of  Distances  Apart  of  Emerging  Axes.  The 
nicols  are  set  at  the  diagonal  positions  and  the  distance  d  from  the 


*May  use  olive  oil,  a  Brom-Napthalin,  Methylene  Iodide,  etc.,  see  p.  95. 

\Fhysikalische  Kryst.,  714. 

t  A.  E.  Tutton,  Philos.  Trans.  R.  Soc.  London,  1894,  185,  913.     E.   A.  Wulfing, 

in.  Mitth.,  1896,  15,  49. 

gTschermak's  Min.  MittheiL,  XIV.,  375. 


OPTICAL  CHARACTERS.  151 

centre  to  either  hyperbola,  or  the  average  of  the  two  distances,  is 

read  by   either  a  glass-  or  screw-micrometer   eye   piece.     Then 

d  d  d 

sin  E  =       »  or  in  liquid,  sin  H—    wv>  or  sin  V  =—,t  in  which 


C  is  a  constant  for  the  same  system  of  lenses  and  is  determined 
once  for  all  with  a  crystal  of  known  axial  angle. 

If  the  section  is  oblique  to  the  bisectrix  it  may  be  tipped 
so  that  the  axis  of  the  microscope  coincides  with  the  direction  of  a 
bisectrix,  and  in  this  way  a  well  defined  interference  figure  is  fre- 
quently obtained.  A  simple  instrument  for  this  purpose*  con- 
sists of  a  clip  supported  15  mm.  above  the  stage  by  a  ball  and 
socket  joint  giving  free  motion  in  all  directions,  through  an  angle 
of  about  45°.  Delicacy  of  manipulation  is  secured  by  a  long  key 
fitting  into  the  clip  arm.  A  special  lens  combination  is  needed  in 
order  to  bring  the  condenser  nearer  the  section. 

(^)  Measurement  by  Rotation  of  the  Section  as  in  tlie  Universal  Ap- 
paratus. This  involves  some  form  of  rotation  apparatus  attached 
to  the  stage  of  the  microscope,  by  which  the  angle  of  rotation  can 
be  read.  Two  forms  of  one  of  the  simpler  devices  are  shown  in 
Fig.  274.  The  crystal  is  fastened  by  wax  to  the  end  of  a  glass 
axle  the  rotation  of  which  is  measured  on  a  graduated  vertical  cir- 
cle. The  crystal  projects  into  a  little  glass  trough  filled  with  re- 
fracting liquid  and  must,  by  trial,  be  adjusted  so  that  the  plane  of 
the'  optic  axes  is  vertical.  The  measured  angle  is  then  2  H.  A 
similar  apparatus  is  described  by  Bertrand.f 

If  the  rotation  necessary  to  bring  the  hyperbolae  into  tangency 
with  the  cross  hair  cannot  be  secured  on  account  of  the  closeness 
of  the  objective  to  the  section  ;  one  hyperbola  may  be  brought  just 
into  the  field,  then  the  other  hyperbola  to  the  corresponding  point 
on  the  opposite  side  of  the  field.  The  angle  of  rotation  necessary 
to  install  the  second  point,  plus  the  angle  equivalent  to  2  d,  the 
distance  apart  of  the  two  points,  will  be  2  H. 

A  much  more  elaborate  device  is  needed  to  both  orient  the 
axial  plane  and  measure  the  axial  angle.  Perhaps  the  best  for  the 
examination  of  small  complete  crystals  is  the  so-called  Universal 
Rotation  Apparatus^  of  Professor  Klein,  Fig.  275,  which  is  espe- 

*Described  by  T.  A.  Jaggar,  Jr.,  Amer.  Jour.  Sci.,  III.,  129,  1897,  and  made  by 
Bausch  &  Lomb  Optical  Co.,  Rochester,  N.  Y. 

\  Hull.  Soc.  Min.  de  France,  III.,  96,  1890. 

£C.  Klein,  Sitzitngsber.  der.  Akad.  Wissenschaften,  Berlin,  1895,  p.  91.  "  Der 
Universaldrehapparat,"  etc. 


152 


CHARACTERS  OF  CRYSTALS. 


FIG.  274. 


cially  designed  as  an  attachment  to  a  No.  VI.  Fuess  Microscope, 

p.  108.  The  microscope  is  tipped 
with  its  axis  in  horizontal  position 
and  the  plate  G  of  the  rotation  ap- 
paratus fastened  to  the  microscope 
stage  by  strong  clips  so  that  the  axle 
P  is  vertical.  The  crystal  is  attached 
at  Fby  wax  and  centered  by  the  cross 
hairs  of  the  microscope  and  the  rect- 
angular and  rotary  movements  of  the 
microscope  stage.  Thereafter  it  may, 
while  still  in  the  line  of  sight,  be 
rotated  measured  amounts  in  three 
planes  at  right  angles  to  each  other 
by  the  vertical  arcs  L  and  Llt  and  the 

horizontal  circle  K.  A  small  cubical  vessel  with  two  opposite  walls 
of  glass  is  filled  with  a  strongly  refracting  liquid  and  raised  into  posi- 
tion between  the  converger  and  objective,  and  so  that  the  crystal  is 
immersed.  If  by  previous  examin- 
ations p.  145  or  from  the  geomet- 
ric properties  any  optical  principal 
section  is  known,  this  will  be 
placed  parallel  to  one  of  the  planes 
of  rotation.  If  by  manipulating 
the  arcs  L  and  L\  with  the  nicols 
horizontal  and  vertical,  a  position  of 
the  crystal  is  found,  such  that  the 
field  remains  dark  throughout  a 
complete  rotation  by  K\  a  prin- 
cipal section  is  horizontal,  for  in 
no  other  sections  would  the  vibra- 
tions of  the  refracted  rays  remain 
horizontal  and  vertical,  p.  1 37.  If 
this  principal  section  is  the  plane  of 
the  optic  axes,  four  times  in  the  course  of  the  rotation  there  will 
be  a  brightening  of  the  field  due  to  the  emergence  of  an  optic 
axis  (internal  conical  refraction),  p.  135.  If  this  does  not  occur 
the  crystal  is  remounted  with  this  principal  section  vertical  and 
parallel  to  one  of  arcs  L,  Lr  Then,  if  the  arc  is  tipped  and  trial 
is  made  by  K,  another  principal  section  must  be  found  and 


FIG.  275. 


OPTICAL  CHARACTERS.  153 

90°  away  the  third,  and  one  of  these  must  be  the  axial  plane. 

Having  found  the  plane  of  the  optic  axes  the  approximate  posi- 
tions of  the  axes  are  noted,  the  converger  is  then  introduced,  the 
nicols  are  set  in  the  diagonal  positions,  the  Bertrand  lens  inserted 
or  the  Klein  ocular,  the  branches  of  the  hyperbola  are  brought  into 
tangency  and  the  four  positions  of  the  emerging  axes  are  read. 

A  special  lens  system  is  needed  with  this  apparatus  to  counter- 
act the  unusual  distance  between  crystal  and  objective. 

MEASUREMENT  OF  THE  TRUE  AXIAL  ANGLE. 

If  the  refracting  liquid  used  in  any  of  the  preceding  methods 
has  an  index  of  refraction  equal  /?  then  sin  H=  sin  V.  To  practi- 
cally obtain  this,  a  liquid  with  n  a  little  above  ft  is  used  and  the  dilu- 
ent added  drop  by  drop  until  the  boundaries  of  the  crystals  fade. 
Then  for  the  accurate  determination  some  coarse  powder  of  the 
substance  is  placed  on  an  object  glass,  a  drop  of  the  liquid  added,  a 
cover  glass  pressed  on  and  the  extinction  directions  determined  in 
a  selected  grain.  With  the  condenser  lowered  and  the  analyzer 
out,  the  two  directions  of  extinction  are  successively  made  to  co- 
incide with  the  vibration  direction  of  the  polarizer,  and  the  micro- 
scope is  focussed  sharply  on  the  dark  boundary  between  the  liquid 
and  the  grain.  The  objective  is  then  raised  and  the  dark  bound- 
ary line  appears  to  move  towards  the  substance  with  the  higher 
index,  of  refraction.  If  the  liquid  is  of  the  mean  refractive  index 
of  the  crystal  the  border  will  for  one  position  move  towards  the 
liquid  and  for  the  other  towards  the  grain. 

A  final  proof  of  the  correctness  of  the  preparation  is  that  if 
«  =  ft  2//a  -f-  2HQ=  1 80°,  but  if  on  measurement  2Ha  +  2//ft< 

I  So0,  then  _<  I    and   too    much    diluent   has   been    added    or  if 
2fia  -f  2//0  <  1 80°  then-  >  I  and  more  diluent  should  be  added. 

CALCULATION  OF  AXIAL  ANGLE  FROM  INDICES. 

This  angle  may  be  calculated  from  the  principal  indices  of  re- 
fraction by  the  equation 


_'  +  ' 

tanF=     «'        "' 


154  CHARACTERS  OF  CRYSTALS. 

DETERMINATION  OF  CHARACTER  OF  RAY  SURFACE. 

This  requires  in  general  a  section  normal  to  a  or  c  that  is  show- 
ing the  lemniscates,  Fig.  266,  with  convergent  monochromatic  light 
or  an  oblique  section  tipped  to  show  these  lemniscates. 

The  difference  between  the  lemniscates  formed  normal  to  the 
acute  bisectrix  and  the  obtuse  bisectrix  cannot  always  be  judged 
and  may  require  measurement  of  the  axial  angle. 

If  the  section  is  normal  to  the  acute  bisectrix  then  the  latter  may 
be  proved  to  be  c  or  a,  that  is,  the  Ray  Surface  proved  to  be  posi- 
tive or  negative  as  follows : 

(a)  The  line  joining  the  optic  axes  is  placed  in  diagonal  posi- 
tion between  crossed  nicols  with  parallel  white  light  the  insertion 
of  a  test  plate  will  determine  as  described,  p.   118,  whether  this 
direction  is  the  vibration  direction  a  of  the  faster  ray  or  c  of  the 
slower  ray  (the  other  extinction  direction  being  6),  that  is,  whether 
this  section  is  normal  to  c  or  a  and  p.  1 36,  Bxa  =  c,  -f-  ;  Bxa  —  a,  — 

(b)  With  convergent  light  and  crossed  nicols  the  line  joining 
the  axial  points  is  placed    in  diagonal   position  and   the    quartz 
wedge,  p.  1 1 8,  gradually  inserted  with  the  direction  c  parallel  to 
this  line.      If  the  crystal  is  positive  the  rings  around  each  axis 
will  expand,  moving  toward  the  centre  and  merging  in  one  curve, 
and  will  contract  and  increase  in  number,  if  the  crystal  be  negative, 
the  change  increasing  with  the  thickness  of  wedge    interposed. 
If  the  direction  c  of  wedge  and  the  line  joining  the  axial  points  are 
crossed  the  effects  are  reversed  ;*  that  is  expansion  of  the  rings 
when  the  length  of  the  wedge  is  parallel  to  the  line  joining  the 
axes  proves  positive  character  and  expansion  when  these  direc- 
tions are  crossed  proves  negative  character. 

(c)  If  with  the  line  joining  the  axial  points  in  normal  position 
the  mica  plate  is  inserted  in  the  diagonal  position  the  black  cross 
is  destroyed  and  the  rings  broken  and  new  flecks  developed  in  the 
same  quadrants  as  in  uniaxial,  but  the  positions  of  the  flecks  do 
not  so  satisfactorily  suggest  the  signs  -f-  and  — 

If  the  section  is  normal  to  the  obtuse  bisectrix  all  the  results  are 
reversed. 
CRYSTALS  IN  THIN  ROCK  SECTIONS. 

The  proper  orientation  of  a  haphazard  section  may  be  obtained 
with  the  Jaggar  apparatus,  but  for  precise  measurement  a  more 

*  Hence  it  is  always  practical  to  develop  the  lemniscate,  even  when  only  the  hyper- 
bolae are  visible,  by  inserting  the  wedge  so  as  to  produce  contraction. 


OPTICAL  CHARACTERS.  155 

elaborate  apparatus  is  essential.  The  different  types  of  univer- 
sal stage  of  von  Federow*  require  the  sections  to  be  mounted 
upon  a  special  round  glass  with  high  index  of  refraction,  while  the 
apparatus  of  Klein,f  Fig.  276,  permits  the  use  of  ordinary  square  or 
rectangular  object  glass,  the  cover  glass,  however,  being  removed. 
The  metal  plate  G  is  clamped  to  the  microscope  stage.  The  metal 
vessel  B,  which  is  somewhat  more  than  a  half  sphere,  is  filled  with 
some  liquid  of  high  refractive  index,  usually  glyerine  n  =  1.46. 

T 


FIG.  276. 

The  light  enters  through  the  glass  plate  a.  The  section  is  sup- 
ported in  the  liquid  by  the  glass  plate  5  in  the  rim  7j  and  re- 
ceives motions  of  rotation  around  the  horizontal  axis  c  by  the  grad- 
uated circle  7]  and  in  its  own  plane  by  the  knob  k,  which  turns  the 
fine-toothed  wheel  z  in  contact  with  teeth  in  the  rim  7\. 

ABSORPTION  AND  PLEOCHROISM. 

The  vibration  directions  of  the  rays  of  monochromatic  light 
which  undergo  greatest  and  least  absorption  and  a  direction  at 
right  angles  to  these  constitute  the  so-called  absorption  axes. 
Their  lengths  are  quite  independent  of  the  lengths  of  the  axes  of 
the  optical  indicatrix,  but  their  directions  are  frequently  though 
not  necessarily  the  same.  Upon  these  axes  can  be  constructed  an 
absorption  ellipsoid  from  which  the  absorption  in  all  directions  can 
be  deduced. 

/;/  otthorliombic  crystals  the  directions  of  the  absorption  axes 
coincide/tfr  all  colors  with  the  directions  of  crystal  axes  and  the 
axes  of  the  optical  indicatrix. 

In  monochnic  crystals  one  absorption  axis  is  parallel  to  the  crys- 
tal axis  b,  therefore  to  one  of  the  axes  of  the  optical  indicatrix;  the 

*  Zeit.f.  Kryst.  v.  25,  p.  351. 

f  Sitzungsber.  Berlin  Akad.,  1895. 


I56 


CHARACTERS  OF  CRYSTALS. 


other  axes  lie  in  the  plane  oio,  but  for  any  color  may  or  may  not 
coincide  with  the  axes  of  the  optical  indicatrix  for  that  color. 

In  triclinic  crystals  no  coincidence  is  essential. 

Although  subject  to  many  exceptions,  the  law  of  Babinet  is 
generally  correct  that  "the  slower  transmitted  ray  is  the  more  ab- 
sorbed." 

The  relative  absorption  of  the  two  rays  for  any  direction  of 
transmission  can  be  obtained  by  use  of  the  spectrophotometer  of 
P.  Glans,  which  consists,  Fig.  277,  of  a  collimator  C,  telescope  H 


FIG.  277. 


FIG.  278. 


and  prism  P.  The  vertical  slit  D  in  the  collimator  is  divided  into 
halves  by  a  cross  piece  of  metal,  and  the  entering  light  passes 
through  a  so-called  Rochon's  quartz  prism  B,  which  gives  a  double 
image,  Fig.  278,  of  each  half  of  the  slit,  the  height  of  the  slit  be- 
ing so  chosen  that  the  extraordinary  image  of  one  and  the  ordinary 
of  the  other  are  in  contact  in  the  centre  of  the  field  and  the  other  two 
are  shut  out  in  the  telescope.  If  now  the  crystal  plate  is  placed  in 
front  of  the  slit  with  its  extinction  directions  parallel  and  at  right 
angles  to  the  slit  the  two  rays  developed  will  pass  through  the 
prism  P  and  their  spectra  be  seen  in  the  telescope  one  above  the 
other,  and  by  means  of  the  horizontally  movable  eye  plate  E  any 
portion  may  be  viewed  alone  and  compared. 

By  introducing  a  nicol  N  arranged  with  a  graduated  circle,  so  that 
when  at  zero  its  plane  of  vibration  coincides  the  principal  section 
of  the  Rochon  prism,  either  ray  may  be  shut  out  at  will  by  turn- 
ing the  nicol  to  o°  or  90°  and  the  spectrum  of  the  transmitted  ray 
compared  with  a  spectrum  scale  sent  through  >S  and  reflected  into 
the  telescope. 

A  quantitative  comparison  can  be  made  as  follows  for  any  portion 
of  the  spectrum.  The  crystal  plate  is  removed  and  the  angle  of 


OPTICAL  CHARACTERS.  157 

rotation  a  of  the  nicol  determined  at  which  the  two  spectra  of  the 
flame  are  equally  bright;  this  will  be  somewhere  near  45°  but 
not  exactly.  The  plate  is  then  replaced  and  the  nicol  again 
turned  through  an  angle  /?  to  secure  equal  intensity  for  the  desired 
portion  of  the  spectrum.  The  relative  intensities  are  : 

f°=  tan2  a  cot2/?.* 

N 

If  only  one  of  the  halves  of  the  slit  is  covered  the  intensity  may 
be  compared  with  that  of  the  incident  light. 

Transmission  along  one  of  the  principal  vibration  directions  a, 
a  or  c  will  yield  the  intensities  for  vibrations  parallel  to  the  other 
two.  As  pointed  out  these  are  directions  of  absorption  axes  in 
the  orthorhombic  crystals  but  not  necessarily  in  the  monoclinic  or 
triclinic  crystals. 

Generally  speaking,  the  records  made  are  simply  comparative, 
for  instance:  Absorption  strong  a>b>c,  or,  Absorption  feeble 
parallel  a.  More  commonly  the  record  is  as  to  pleochroism  or 
effect  with  white  light,  the  color  differences  for  vibrations  parallel 
b,  b  and  c  for  some  middle  color  being  recorded,  or,  the  colors 
for  directions  referred  to  some  definite  crystal  face.f 

The  pleochroic  images  are  viewed  either  with  a  microscope  or 
dichroscope  as  described,  p.  131,  or  the  apparatus  of  Klein,  Fig. 
275,  may  be  conveniently  used,  after  the  measurement  of  the  angle 
between  the  optic  axes,  by  adjusting  the  crystal  with  the  plane  of 
the  optic  axes  horizontal.  If  the  plane  of  vibration  of  the  polarizer 
is  vertical  and  the  analyzer  is  out,  the  color  obtained  throughout 
a  rotation  by  K  corresponds  to  b.  If  the  vibration  plane  of  the 
the  polarizer  is  horizontal,  transmission  parallel  to  the  bisector  of 
axial  angle  yields  the  color  for  one  of  a  or  c  and  by  a  rotation  of 
90°  with  K  the  color  of  the  other  results. 


*Liebisch  Grundriss  der  Phys.  ICryst.,  p.  312. 

f  For  example:  Andalusite.  Pleochroism  a  blood  red  to  rose  red,  b  =  c  olive 
green.  Dana's  System,  p.  496. 

Titanite.  a  =  yellowish  red,  6  —  greenish  red,  c  =  pale  yellow.  Ripidolite.  Yellow 
to  reddish  for  vibrations  normal  to  coi,  green  parallel  ooi.  Rosenbusch  liuifitabellcn, 
Hie. 

Axinite.  Pieochroism  strong,  normal  to  r  pale  olive-green,  giving  with  dichroscope 
olive-green  and  violet  blue,  parallel  r  and  normal  to  edge  r\M  cinnamon  brown,  giving 
cinnamon  brown  and  violet  blue.  Dana's  System,  p.  528. 


158 


CHARACTERS  OF  CRYSTALS. 


FIG.  279. 


ABSORPTION  TUFTS. 

Plates  of  crystals  with  strong  absorption,  especially  if  cut  normal 
to  an  optic  axis,  often  show  in  convergent  non-polarized  light  two 
dark  tufts  symmetrical  to  the  plane  of  the  optic  axes,  but  sepa- 
rated by  a  bright  space.  If  white  light  is  used  the  pencils  are  of 
different  color  to  the  rest  of  the  field. 

In  Fig.  279,^  is  the  optic  axes  to  which  the  plate  is  normal ;  A' 

is  the  other  optic  axis.  Oblique  rays 
emerging  at  any  point  m  on  the  line 
AAf  have  been  transmitted  in  the 
plane  of  the  optic  axes  and,  there- 
fore, one  of  the  rays  vibrates  in  the 
plane  of  the  optic  axis,  the  other 
parallel  to  the  principal  vibration  dir- 
ection b  or  AN.  Oblique  rays  from 
any  point  m'  on  the  line  AN  normal 
to  A  A'  will  vibrate  (see  p.  1 37)  par- 
allel and  at  right  angles  to  the  bi- 
sector of  the  angle  Am'A'.  As 
explained  by  Mallardf  the  intensity 
of  the  pencil  of  rays  emerging  at  m1  is  necessarily  less  than  that  of 
the  pencil  of  rays  emerging  at  m,  the  difference  increasing  the 
nearer  the  points  are  to  each  other,  that  is,  normal  to  AA'  is  a  dark 
tuft  growing  lighter  on  each  side  of  A.  Similar  effects  are  ob- 
tained for  points  not  on  AN,  the  total  result  being  two  tufts  or 
pencils  of  hyperbolic  shape  tangent  to  AA'. 

When  white  light  is  used  if  the  optic  axes  are  dispersed,  the  tufts 
are  not  exactly  superposed  and  the  borders  show  color  tints. 

The  test  may  be  made  by  holding  the  crystal  section  close  to 
the  eye  and  looking  at  the  bright  sky  or  by  removing  the  polarizers 
from  a  converging  polariscope. 

If  the  plates  are  examined  in  convergent  polarized  light,  the 
analyzer  being  removed,  two  types  may  be  distinguished  as  follows  : 
TYPE  I.  The  dark  tufts  will  appear  normal  to  the  plane  of  the 
optic  axis  and  in  a  clear  field,  when  the  plane  of  the  optic  axis 
is  normal  to  the  vibration  plane  of  the  polarizer.  Example :  Anda- 
lusite  of  Brazil.  Plate  cut  normal  to  acute  bisectrix  and  placed 
on  a  thin  plate  of  azurite  shows  black  tufts  on  a  blue  ground. 

*  W.  Voigt  zur  Theorie  der  Absorption  des  Lichtes  in  Krystallen  IViedemann's 
Ann.,  1885,  XXIII.,  577. 

f  Traiet  de  Cristallographie,  II.,  361. 


OPTICAL  CHARACTERS.  159 

TYPE  II.  The  same  phenomenon  results  when  these  planes  are 
parallel.  Example :  Epidote  of  Knappemvand  Tyrol.  Cleavage 
plate  parallel  ooi  gives  tufts  symmetrical  to  oio,  with  white  light 
the  borders  towards  the  acute  bisectrix  are  dark  green  and  the 
opposite  are  dark  red. 

In  both  types,  with  a  rotation  of  the  plate  through  90°  from 
these  positions,  tufts  also  appear  parallel  to  the  plane  of  the  optic 
axes. 

METALLIC  REFLECTIONS  OR  METALLIC  LUSTRE. 

The  metals  do  not  obey  the  usual  laws  of  reflection ;  incident 
plane  polarized  light  is  reflected  as  such,  only  when  the  plane  of 
vibration  of  the  incident  light  is  parallel  or  at  right  angles  to  the 
plane  of  reflection ;  and  in  both  of  these  cases  there  is  developed 
a  change  of  phase,  which  is,  however,  greater  in  the  latter  case 
than  in  the  former.  For  all  other  azimuths  of  vibration  the  vibra- 
tion of  the  reflected  light  is  the  result  of  two  unequal*  compo- 
nents at  right  angles  and  is,  therefore,  elliptically  polarized. 

The  intensity!  of  the  reflected  light  differs  also  from  that  obtained 
with  transparent  substances.  For  normal  incidence  it  is  not  zero, 
but  considerable,  decreasing  as  the  angle  of  incidence  increases  up 
to  a  value  of  50°  to  60°,  thereafter  again  increasing  until  the 
angle  of  grazing  incidence  is  reached. 

,  For  a  given  incidence  certain  colors  are  reflected  by  metals 
more  strongly  than  others,  whereas,  transparent  substances,  even 
if  colored,  reflect  white  light.  The  combined  effect  of  this  intense 
and  selective  reflection  is  called  Metallic  Lustre. 

SURFACE  COLORS. 

Between  metallic  and  transparent  substances  are  certain  which 
are  transparent  for  light  of  certain  wave-lengths,  but  stop  others  so 
completely  that  even  with  thin  sections  they  are  represented  in 


*Varying  from  o  for  normal  incidence  to  A/2  for  grazing  incidence  and  for  a  parti- 
cular angle,  which  is  experimentally  found  to  be  the  angle  of  maximum  polarization  of 
common  light,  passing  through  the  value  A/4. 

f  Cauchy,  Voigt  and  Drude  have  studied  the  phenomena  and  calculated  the  indices 
of  refraction  and  absorption  for  a  number  of  metallic  substances  for  different  vibration 
directions  upon  fresh  cleavage  surfaces.  For  gold  and  silver  a  higher  velocity  c  flight 
was  found  than  in  air,  but  very  high  absorption ;  for  lead,  platinum  and  iron  the  indices 
of  refraction  were  high,  but  the  absorption  relatively  low.  Very  high  absorption  indices 
were  obtained  for  stibnite  and  galena.  Kundt  discovered  phenomena  of  double  re- 
fraction of  light  transmitted  through  extremely  thin  electrically  deposited  metallic  films. 


160  CHARACTERS  OF  CRYSTALS. 

the  spectrum*  by  absorption  bands.  These  rejected  colors  are 
reflected. 

When  white  light  is  used  the  reflected  or  surface  color  is  strik- 
ingly unlike  the  transmitted  or  body  color,  for  instance,  a  drop  of 
fuchsin  solution  dried  on  a  glass  plate  shows  the  surface  color 
green  with  metallic  lustre  and  the  body  color  red,  or  if  a  glass  be 
laid  over  the  surface  color  is  blue-green.  A  crystal  of  potassium 
permanganate  polished  to  a  thin  layer  on  glass  gives  a  yellow  sur- 
face color  and  a  dark  violet  body  color. 

The  surface  and  body  colors  are  only  approximately  comple- 
mentary as  Wiedemannf  has  shown.  The  surface  colors  in  uniaxial 
crystals,  moreover,  are  non-polarized  when  reflected  from  basal 
sections,  but  dichroic,  that  is,  elliptically  polarized  and  made  up  of 
two  components  vibrating  in  and  at  right  angles  to  the  plane  of 
reflection  for  other  sections.  For  instance,  with  crystals  of  mag- 
nesium platinocyanide  the  light  reflected  from  the  basal  plane  is 
violet  and  not  to  be  decomposed,  whereas  the  green  color  from  a 
prism  face  is  always  resolvable  into  two  colors  varying  with  the 
angle  of  incidence. 

FLUORESCENCE. 

Certain  organic  compounds,  fluorite  containing  dissolved  organic 
matter  and  uranium  glass  possess  the  property  of  absorbing  light 
and  again  emitting  it  as  light  of  a  different  color.  If  sunlight  is 
focussed  on  a  dilute  solution  of  quinine  sulphate  it  becomes  deep 
blue  near  the  surface  of  entrance.  If  the  spectrum  of  the  trans- 
mitted light  is  examined  the  violet  and  ultra  violet  rays  are  missing, 
that  is  these  have  in  the  solution  been  turned  into  blue.  A  test 
tube  of  the  solution  held  in  a  solar  spectrum  beyond  the  visible 
rays  becomes  blue. 

In  fluorite  and  uranium  glass,  which  are  isotropic,  the  color 
produced  is  independent  of  the  direction  of  transmission.  In  the 
uniaxial  magnesium  platinocyanide  crystals^  with  sunlight  passed 
through  blue  or  violet  glass  the  green  surface  color  disappears  on 
the  prism  faces,  and  by  polarizing  either  the  incident  or  emerging 
light  a  dichroic  fluorescence  is  observed,  scarlet  red  for  vibrations 

^Christiansen  found  that  the  spectrum  of  the  red  solution  of  fuchsin  lacked  green 
and  that  blue  was  deviated  more  than  red. 
•\Ann.  de  Phys.  v.  151,  p.  625,  1874. 
{Lommelin  Ann.  de  Fhys.,  VIII.,  634,  XLIV.,  311. 


OPTICAL  CHARACTERS.  161 

normal  to  the  optic  axis,  orange  yellow  for  vibrations  parallel  to 
the  optic  axis.  If  the  violet  rays  are  incident  normal  to  the  base, 
polarization  reveals  only  one  color,  orange  yellow,  whatever  the 
azimuth  of  the  vibrations. 

PHOSPHORESCENCE. 

Phosphorescence,  or  the  emission  of  light  by  a  substance  in  the 
dark  after  it  has  been  exposed  to  light,  heat,  friction,  mechanical 
force,  or  an  electrical  discharge,  has  not  been  clearly  shown  to  be  de- 
pendent in  any  way  upon  the  crystalline  structure.  It  is  developed 
by  one  process  or  another  in  a  large  number  of  minerals  and  salts.* 

THE    NORREMBERG   AND    REUSCH    COMBINATIONS    OF  MlCA    PLATES. 

The  optical  phenomena  of  combinations  of  differently  oriented 
layers  of  a  doubly-refracting  substance  are  of  great  value  in  the 
development  of  theories  of  structure.  Muscovite  in  which  the 
acute  bisectrix  is  nearly  normal  to  the  cleavage  plane  is  especially 
convenient  for  these  experiments. 

If  two  plates  of  muscovite  of  equal  thickness  are  laid  so  that 
their  axial  planes  are  at  right  angles,  then  with  parallel  light  and 
crossed  nicols,  the  phase  difference  originated  in  one  is  exactly 
compensated  by  that  in  the  other,  and  the  field  will  be  dark 
throughout  rotation  of  the  stage.  With  convergent  light  four  axes 
will  emerge  equally  distant  from  the  centre  and  between  them 
thin  hyperbolic  color  curves,  the  asymptotes  to  which  form  a  black 
cross. 

Norremberg  discovered^  that  if  the  plates  were  made  so  thin 
that  the  phase  difference  of  the  rays  developed  in  any  plate  was 
less  than  a  wave-length,  and  that  if  a  pile  of  12  to  36  of  these  were 
made  the  axial  planes  in  adjacent  plates  being  at  right  angles,  then 
not  only  did  the  compensation  of  the  phase  differences  give  a  dark 
field  with  parallel  light  and  crossed  nicols,  but  with  convergent 
light  the  isochromatic  curves  produced  by  the  oblique  rays  were 
circles,  that  is,  the  effect  of  the  pile  was  exactly  equivalent  to  a 
basal  section  of  an  uniaxial  crystal,  Fig.  280. 

Reusch  found,J  by  piling  12  to  36  similar  plates  so  that  the  axial 
planes  of  adjacent  plates  were  at  120°  to  each  other  as  in  Fig.  281, 

*Lommel  in  Ann.  de  P/iys.,  VIII.,  634,  XL1V.,  311. 

|D.  Hahn,  Phosphorescenz  der  Mineralien,  Zeif.  f.  d.  ges.  Natur-wiss.,  Bd. 
XLIIL,  I,  131,  1874. 

\  Berichte.  Ak.  Wissens.  Berlin,  July  8,  1869,  p.  530. 


1 62 


CHARACTERS  OF  CRYSTALS. 


that  with  parallel  incident  light  there  emerged  a  plane  polarized 
ray,  the  vibration  plane  of  which  had  been  rotated  in  the  direc- 
tion of  the  piling  through  an  angle  proportionate  to  the  thick - 


FIG.  280. 


FIG.  281. 


ness  of  the  pile  that  is,  if  starting  at  the  lowest  plate  the  piling 
had  been  with  a  change  in  the  direction  of  the  hands  of  watch  the 
rotation  was  also  in  that  direction,  and  the  color  produced  with 
white  light  would,  by  rotation  of  the  analyzer,  also  in  the  direc- 
tion of  the  hands  of  a  watch,  pass  in  the  order,  red,  orange,  yellow, 
green,  blue,  violet. 

With  convergent  light  Fig.  281  resulted,  the  centre  being  colored. 

This  conforms  strictly  to  the  behavior  of  a  basal  plate  of  right- 
handed  quartz,  except  that  by  rotation  of  the  stage,  a  slight  change 
of  color  takes  place,  this,  however,  decreases  as  the  mica  plates 
are  made  thinner. 

If  the  piling  were  in  opposite  direction  the  rotation  of  the  plane 
of  vibration  would  be  opposite  and  the  color  order  also.  Based  on 
these  and  similar  experiments,  theories  as  to  the  structure  of  cer- 
tain crystals  have  been  developed  by  Mallard,*  Sohncke  and  others. 

*  Trait*  de  Cristallographie  II.,  262-304;  Zeit.f.  Kryst.,  etc.,  XIX.,  529. 


PART  III,  THE  THERMAL,  MAGNETIC  AND  ELECTRL 

CAL   CHARACTERS,  AND   THE   CHARACTERS 

DEPENDENT  UPON  ELASTICITY 

AND  COHESION. 


CHAPTER  XII. 


THE  THERMAL  CHARACTERS. 


Certain  substances,  such  as  halite,  are  essentially  transparent  to 
both  heat  and  light ;  and  if  sunlight  be  decomposed  by  a  prism  of 
such  a  substance  there  is  obtained  not  only  the  visible  color  spec- 
trum consisting  of  the  rays  between  the  least  refracted  red  (A  = 
760.4  /J.IJL)  and  the  most  refracted  violet  (A=  393.3  ////),  but  if  each 
ray  is  tested  by  a  delicate  thermo-pile  the  temperature  will  be  found 
to  increase  from  the  violet  end  towards  the  red  and  to  a  certain  dis- 
tance beyond  the  red  where  it  is  a  maximum  and  thereafter  to  de- 
crease, proving  the  existence  of  invisible  heat  rays  beyond  the  red. 

Experiments  show  also  that  these  heat  rays  are  reflected  and 
refracted  and  absorbed  like  light  rays,  that  they  may  be  doubly 
refracted,  as  first  shown  by  Knoblauch  in  calcite,  that  they  may  be 
polarized  by  reflection  or  refraction  and  that  when  polarized  their 
power  to  penetrate  crystals  varies  not  only  with  the  direction  of 
transmission  but  with  the  direction  of  vibration.  For  instance,  heat 
will  not  penetrate  crossed  nicols,  but  the  interposition  of  a  plate  of 
doubly-refracting  substance  will  permit  components  to  penetrate, 
except  at  intervals  of  90°  when  the  planes  of  vibration  of  the  heat 
rays  produced  in  the  plate  coincide  with  those  of  the  nicols.  In- 
terference and  circular  polarization  of  heat  rays  have  also  been 
experimentally  proved. 

Heat  rays  may,  therefore,  be  regarded  as  invisible  light  rays 
which  are  in  general  of  greater  wave-length  and  less  refrangibility 
than  the  visible  rays,  but  are  subject  to  the  same  laws.  It  is  pos- 
sible, though  difficult,  to  determine  experimentally  a  series  of  con- 
stants for  crystals  with  respect  to  these  invisible  rays,  as  in  the 
famous  experiments  of  Knoblauch.* 

*  Knoblauch,  Pogg.  Ann.,  LXXXV,  169;  XCIII,  161. 


1 64  CHARACTERS  OF  CHVSTALS. 

Simpler  and  more  convenient  tests  can  be  made  with  respect  to 
the  conductivity  or  rate  of  transmission  of  heat  from  particle  to 
particle,  and  to  expansion.  For  both  a  dependence  is  found  to 
exist  upon  the  crystal  structure. 

HEAT  CONDUCTIVITY. 

The  shape  of  the  "  surface  of  conductivity  "  or  isothermal  surface, 
the  radii  of  which  are  the  rates  of  conductivity  in  different  direc- 
tions, is  most  easily  determined  from  the  surface  conductivities  in 
sections  of  known  orientation. 

The  relative  surface  conductivies  are  most  satisfactorily  obtained 
by  Rontgen's  modification  of  the  older  de  Senarmont  method. 
The  previously  cleaned  and  polished  section  is  breathed  upon, 
quickly  touched  by  a  very  hot  metal  point  normally  applied,  and 
instantly  dusted  with  lycopodium  powder.  The  section  is  then 
turned  upside  down  and  tapped  carefully,  when  the  powder  falls 
from  where  the  moisture  film  had  been  evaporated,  but  adheres 
elsewhere,  giving  a  sharply  outlined  figure.  The  entire  operation 
should  take  less  than  three  seconds. 

In  all  homogeneous  substances  in  which  the  surface  tested  is 
large  enough  to  allow  for  irregularity  of  contour  the  resultant  fig- 
ure is  either  an  ellipse  or  a  circle,  that  is  a  section  of  an  ellipsoid. 

The  major  and  minor  axes  of  the  ellipse  are  carefully  measured 
under  the  microscope  with  a  micrometer  eyepiece. 

In  the  old  method  of  de  Senarmont  the  surface  of  the  section  is 
coated  with  a  very  thin  layer  of  wax  or  paraffin  either  applied  with 
camels'  hair  brush  *  or  by  melting  a  thin  bit  on  the  surface  and 
pouring  off  the  excess.  The  wax-covered  surface  is  placed  in  a 
horizontal  position  and  heat  applied  at  one  point  either : 

(a)  By  a  wire  or  narrow  tube  of  metal  (silver,  platinum,  copper) 
passing  through  a  hole  pierced  normal  to  the  waxed  surface  and 
heated  either  by  a  lamp  at  some  distince,f  or  by  an  electric  current, J 
or  by  a  current  of  hot  air.  § 

(&)  By  touching  with  a  hot  pointed  wire  vertically  applied. 

(c)  By  a  little  platinum  ball  soldered  to  a  platinum  wire,  the  latter 
being  surrounded  to  avoid  radiation  and  heated  by  an  electric  cur- 
rent. 


*Tyndall,  Heat  as  a  Mode  of  Motion,  p.  189,  D.  Appleton,  1891. 

f  de  Senarmont,  1847,  Ann.  de  Chitn.  et  de  phys.,  XXI.,  457;  XXII.,  179. 

JTyndall,  1.  c.,  190. 

gjannetaz,  Bull.  toe.  Min.,  I.,  19. 


THERMAL  CHARACTERS.  165 

Voigt  recommends*  a  mixture  of  3  parts  elaidic  acid  with  I 
part  of  wax,  the  mixture  melting  at  40°  to  50°  C. 

A  constant  temperature  is  maintained  until  the  coating  has 
melted  far  enough  from  the  point  of  application  of  the  heat.  The 
boundary  of  the  melted  patch  is  then  the  isothermal  curve  show- 
ing the  distances  which  the  heat  has  been  transmitted,  and  is 
visible  after  cooling  as  a  ridge  which  is  always  in  the  shape  of 
an  ellipse  or  circle. 

As  a  result  of  the  experiments  it  is  found  that : 

i°  In  singly-refracting  crystals  (isometric)  all  sections  yield 
circular  isothermal  curves;  that  is,  the  isothermal  surface  is  a  sphere. 

2°  In  optically-uniaxial  crystals  ( tetragonal  or  hexagonal  )  ba- 
sal sections  yield  circular  curves,  but  all  other  sections  yield  ellip- 
tical curves  which  become  more  eccentric  as  the  section  becomes 
more  nearly  paraiiei  to  the  optic  axis.  That  is,  the  isothermal  sur- 
face is  an  ellipsoid  of  revolution  around  c ,  the  optic  axis.  As  for 
light  a  division  may  be  made  into  -f  and  — .  For  instance,  in 
quartz  the  conductivity  is  more  rapid  parallel  c  and  in  calcite  at 
right  angles  to  c,  hence  quartz  is  thermally  positive,  calcite 
thermally  negative. 

3.  In  optically-biaxial  crystals  the  isothermal  curves  are  all 
ellipses  and  the  isothermal  surface  is  a  triaxial  ellipsoid,  the  axes 
of  which  are  at  right  angles  to  each  other.  In  orthorhombic 
crystals  these  axes  coincide  with  the  crystallographic  axes,  and 
the  major  and  minor  axes  of  the  ellipse  obtained  in  any  principal 
section  are  parallel  to  the  crystal  axes  therein.  In  monoclinic 
crystals  one  axis  of  the  isothermal  surface  coincides  with  b  and 
the  other  two  lie  in  the  plane  oio.  In  the  zone  [oio  100]  the 
major  and  minor  axes  of  the  isothermal  curves  should  coincide 
with  crystal  axes.  In  triclinic  crystals  there  is  no  essential  or 
probable  parallelism  of  crystal  axes  and  axes  of  ellipses  in  any 
section.t 

*  Elemente  der  Krystallphysik,  78. 

f  Jannetaz  showed  an  apparent  connection  between  directions  of  cleavage  and  ease 
of  conductivity.  For  instance,  the  isothermal  surface  for  uniaxial  crystals  which  have 
an  easy  basal  cleavage  is  usually  a  flat  oblate  spheroid.  If  the  direction  of  cleavage  is 
not  basal,  but  oblique,  the  longer  axis  will  often  be  in  the  section  most  nearly  parallel 
to  the  cleavage.  As  will  be  later  explained,  planes  of  cleavage  are  supposed  to  be 
directions  of  greatest  closeness  of  molecules,  and  it  seems,  therefore,  that  the  rapidity  of 
transmission  increases  as  the  molecules  are  closer  together.  It  is  curious,  however, 
that  if  an  amorphous  substance,  such  as  glass,  is  subjected  to  compression  or  tension  the 
isothermal  circles  become  elliptical,  the  shorter  axis  being  in  the  line  of  greater  press- 
ure or  packing  of  molecules. 


1 66  CHARACTERS  OF  CRYSTALS. 


EXPANSION  BY  HEAT. 

Most  *  substances  increase  in  volume  when  heated  and  contract 
on  cooling,  and  experiments  show  that  in  crystals  the  rate  of  ex- 
pansion is  not  the  same  for  directions  crystallographically  unlike, 
but  varies  as  follows  : 

1°  Isometric  Crystals.  The  rate  of  expansion  is  the  same  for  all 
directions.  A  sphere  heated  becomes  a  sphere  of  larger  diameter. 

2°  Tetragonal  and  Hexagonal  Crystals.  The  rate  of  expansion 
is  the  same  for  all  directions  equally  inclined  to  the  axis  r,  is  either 
a  maximum  or  a  minimum  parallel  to  c,  and  varies  regularly  from 
this  to  the  direction  at  right  angles.  A  sphere  heated  becomes  an 
ellipsoid  of  revolution  around  c  as  an  axis. 

3°  Orthorhombic,  Monoclinic  and  Iriclinic  Crystals.  There  is  no 
axis  of  isotropy.  The  directions  of  maximum  and  minimum  expan- 
sion are  at  right  angles,  and  a  sphere  heated  becomes  a  triaxial  ellip- 
soid, the  axes  being  these  directions  and  a  third  at  right  angles  to 
their  plane. 

Orthorhombic,  the  axes  coincide  with  the  crystal  axes. 

Monoclinic,  one  axis  coincides  with  b,  the  others  lie  in  oio,  but  not 
coincident  with  conductivity  axes  or  principal  vibration  directions. 

Triclinic,  the  axes  have  no  fixed  positions  relative  to  the  crystal 
axes. 

DIRECT  MEASUREMENT  OF  LINEAR  EXPANSION. 

The  coefficient  of  linear  expansion,  that  is  the  increase  in  a 
unit  of  length  for  a  temperature  change  from  o°  to  i°  C.  may  be 

accurately  measured  for  any  direction 
by  the  method  of  Fizeau  f  which  as 
perfected  by  Abbe  is  as  follows :  J 

In  Fig.  282  0  is  a  plane  parallel 
plate  of  the  crystal,  about  10  mm. 
thick,  which  rests  upon  three  projec- 
tions turned  upon  the  steel  plate  7. 
Three  screws,  with  a  fine  2  mm.  thread, 
pass  through  T  and  support  the  glass 
plate  P,  which  tapers  slightly  so  that 

*A  few  expand  on  cooling  below  a  certain  point  and  one,  at  least,  iodide  of  silver, 
contracts  when  heated  above  the  ordinary  room  temperatures. 

f  Description  by  J.  R.  Benoit,  Trans,  et  Mem.  du  bur.  internat.  des  poids  et 
wes.,1.,  i,  1881;  VI.,  i.  1888. 

\  Described  by  C.  Pulfrich,  Zeit.  f.  Instrumentkunde,  XIII.,  365,  401,  437. 


THERMAL  CHARACTERS. 


167 


when  the  upper  surface  is  horizontal  the  lower  is  inclined  about 
20  minutes.  By  adjusting  the  screws  a  thin  wedge-like  film  of  air 
is  left  between  the  horizontal  polished  surface  of  the  substance  and 
the  lower  surface  of  the  glass. 

The  telescope  PO,  Fig.  283,  is  at  once  telescope  and  collimator, 
that  is  the  light  from  a  Geissler  tube,  shown  at  L  in  the  smaller 


Fio.  283. 

figure,  enters  at  the  side,  is  deflected  by  the  prism  P,  made  parallel 
by  the  lens  system  O,  decomposed  by  the  two  flint  glass  prisms  Pl 
and  P2  and  by  varying  the  angles  at  which  these  are  set  by  the  screw 
S,  rays  of  any  chosen  color  are  made  to  pass  vertically  through  R 
and  reach  the  interference  apparatus  within  the  brass  vessel  G. 

The  rays  incident  at  the  upper  surface  of  the  crystal  plate  and  the 
lower  surface  of  the  glass  plate  interfere  on  reflection  producing 
parallel  dark  bands  wherever  the  thickness  of  the  air  film  is  -J-A,  f  A, 
etc.  The  distance  between  two  adjacent  bands  is  therefore  a  func- 
tion of  the  wave-length  of  the  light  used.  This  distance  is  meas- 
ured by  a  screw  micrometer  M  which  moves  a  vertical  double 
hair  horizontally  across  the  held. 

The  vessel  G  containing  the  interference  apparatus  is  enclosed 
in  two  concentric  cylinders,  the  outer  one  containing  a  liquid. 
When  the  liquid  is  heated  the  crystal  plate  and  the  interference 


1 68 


CHARACTERS  OF  CRYSTALS. 


apparatus  both  expand  and  the  interference  bands  change  in  dis- 
tance apart.  From  this  change  and  from  the  previously  deter- 
mined effect  of  the  expansion  of  the  apparatus  the  change  due  to 
the  expansion  of  the  crystal  is  calculated.* 

Voigt  describesf  a  simpler  apparatus  in  which  the  expansion 
of  the  crystal  tips  one  of  two  parallel  mirrors.  Reflected  signals 
from  the  two  mirrors  are  viewed  by  a  telescope  one  meter  distance 
and  their  divergence  measured.  The  apparatus  is  heated  by  im- 
mersion in  paraffin  oil  at  60°  to  70°  C. 

MEASUREMENT  OF  EXPANSION  BY  CHANGE  IN  DIEDRAL  ANGLES. 

The  indices  and  therefore  the  zone  rela- 
tions are  not  changed  by  uniformly  heat- 
ing a  crystal,  for  any  series  of  points  on  a 
straight  line  remains  on  a  straight  line  and 
at  the  same  proportionate  distances  apart. 
If,  therefore,  in  Fig.  284  ABC  is  the  unit 
plane  and  HKL  any  other  plane,  the  points 
0,  K  and  B  will  after  expansion  be  on  a 
straight  line  and  the  same  proportionate 
distance  apart,  so  also  0,  L  and  C  or  0,  H 
and  A.  Therefore,  the  indices  of  HKL 
after  expansion  will  be 

t      OL'       OL 

^__    t>    • 

OB 


FIG.  284. 


QA'       OA 


OHJ~OH 


OK'      OK  _ 
'  ~         ~k; 


OB' 


In  isometric  crystals  the  angles  are  unchanged  by  expansion,, 
but  in  all  other  systems  all  diedral  angles  are  altered  except  that : 

(a)  Faces  parallel  to  two  expansion  axes  remain  parallel  to  their 
original  position  and  normal  to  all  faces  in  the  zone  of  the  third  axis. 

(&)  If  the  rates  of  expansion  are  equal  parallel  to  two  axes  the 
faces  in  the  zone  of  the  third  axis  remain  parallel  to  their  original 
positions. 

In  tetragonal  and  hexagonal  crystals  by  (a)  the  basal  planes 
remain  at  90°  to  the  faces  in  the  prismatic  zone,  so  also  any 
face  in  the  prism  zone  remains  at  90°  to  all  faces  of  the  zone 
originally  normal  to  it  and  by  (b)  all  angles  in  the  prismatic  zone 
remain  constant. 


*Zeit.f.  Instk.,  XIII.,  440,  etc. 
\  Wied.  Ann.,  XLIIL,  831,  1891. 


THERMAL  CHARACTERS.  169 

In  orthorhombic  crystals  since  a,  b  and  c  are  expansion  axes,  each 
pinacoid  remains  at  90°  to  every  face  in  the  zone  of  the  third 
axis. 

In  monoclinic  crystals  the  clino  pinacoid  remains  at  90°  to  faces 
in  the  zone  of  b. 

From  the  change  in  angle  the  linear  coefficients  may  be  calcu- 
lated.*    

The  change  in  angle  is  usually  small,      /~  /  / 

requiring  delicate  measurement.  It  may 
be  demonstrated  f  by  cutting  a  plate - 
like  cleavage  of  calcite,  Fig.  285  a,  / \  \ 

normally  and  parallel  to  the  longer  di-    ^- ^ ^ 

agonal  of  the  larger  face,  reversing  one- 
half  and  gluing  as  in  b  with  water  glass. 
On  heating,  the  adjacent  angles  expand 
in  the  same  direction  and,  as  exagger- 
ated in  c,  one-half  becomes  inclined  to 
the  other  about  20'  for  100°  C.  temperature  change.  This  bend- 
ing is  easily  shown  by  reflection  of  a  signal  on  a  screen  3  meters 
distant. 


DETERMINATION  OF   EXPANSION  BY  CHANGES  IN  THE  OPTICAL 

PHENOMENA. 

The  expansion  of  a  crystal  produces  a  change  in  the  rapidity  of 
transmission  of  light  which  may  be  determined  by  measurement  of 
the  indices  of  refraction  or  indirectly  by  the  interference  phe- 
nomena. 

In  singly  refracting  (isometric)  crystals  the  change  is  alike  in  all 
directions.  The  crystal  remains  optically  isotropic  but  the  index 
of  refraction  may  become  smaller  as  with  fluorite,  or  larger,  as  with 
diamond. 

In  uniaxial  (tetragonal  and  hexagonal)  crystals  the  changes  in 
velocity  are  unequal,  that  is  the  principal  indices  of  refraction  and 

*  For  instance  if  the  rhombohedral  angle  of  a  calcite  cleavage,  which  decreases  almost 
9'  for  100°  elevation,  is  105°  5'  at  10°  C.  and  104°  56'  at  110°  C.  then  by  formulae, 
sin  a  =  cosi^X  I-I55>  tan  a  X  .866  =  ^,  (Moses*  and  Parson's  Mineralogy,  p.  60)  we 
have  /  sin  a  =  9.84663,  a  =  44°.  63,  tan  a  X  -866  =  c  —  .8549,  and  /  sin  a'  ir=  9.84732, 
a'  =  44°  72,  tan  a  X  -866  =  c'  =  -85/ 5- 

f  Voigt,  Elemente  der  Krystalphysik,  p.  49. 


CHARACTERS  OF  CRYSTALS. 


a  vary  unequally  and  not  necessarily  in  the  same  direction.*  The 
strength  of  the  double  refraction  ?  —  «  may  be  either  increased,  rais- 
ing the  interference  color  and  contracting  the  rings  in  the  interfer- 
ence figure,  or  the  reverse  may  take  place.  The  change  when  y  —  a 
is  made  less  may,  for  a  certain  temperature,  reduce  the  double  re- 
fraction to  zero,  that  is  for  that  color  of  light  and  at  that  tempera- 
ture the  crystal  is  optically  isotropic  and  for  a  further  temperature 
change  in  the  same  direction  will  change  in  optical  character 
from  +  to  —  or  —  to  -f . 

In  biaxial  crystals  (orthorhombic,  monoclinic  and  triclinic)  the 
unequal  changes  in  a,  /§  and  y,  the  principal  indices  of  refraction, 
are  indicated  not  only  by  raising  or  lowering  the  interference  colors 
and  in  the  contraction  or  expansion  of  the  rings  of  the  interference 
figure  but  by  changes  in  the  angle  between  the  optic  axes,  which 
is  simply  a  function  of  the  principal  indices.  If  either  a  or  f 

at  any  temperature  become  equal  to 
/5  then  for  that  temperature  and  light 
the  crystal  is  uniaxial  and  the  axial 
angle  is  zero  and  for  a  further  change 
in  the  same  direction  the  former 
middle  index  ft  will  become  «  or  ?t  that  is  b  will  become  a  or  c  and 
the  optic  axes  will  pass  into  a  plane  at  right  angles  to  the  former 
position  of  the  axial  plane.  Fig.  286  represents  successive  changes 
in  such  a  transformation. 

The  following  selected  examples  illustrate  these  facts. 

BARITE  (Orthorhombic).  2.E  for  sodium  flame  increases  with  rising  temperature 
from  64°  i'  at  20°  C.  to  68°  51'  at  100°  C.  and  77°  16'  at  200°  C. 

GLAUBERITE  (Monoclinic).  The  principal  vibration  directions  are  essentially  un- 
changed, but  2.E  decreases  with  rising  temperature.  Within  a  range  of  40°  C.  the  crys- 
tals become  successively  uniaxial  for  all  colors  and  the  axes  thereafter  are  in  a  plane  at 
right  angles  to  the  former  axial  plane. 


FIG.  286. 


Temperature 
Centigrade. 
1  8° 

22° 

36° 
46° 
58° 

Apparent  Angle  2E   for 
Li  Red.            Na  Yellow.              Tl  Green. 

13°  30'                 11°    8'                   8°  14' 
11°    i'                 8°    9'                  o° 
8°  40'                  o°                        7°    8' 
o°                         7°  14'                10°  32' 

Blue. 

0° 

8°  42' 
11°    8' 

13°     2 

*  With  increased 

temperature  : 

Calcite 

a 
increased 

7 
increased 

7  —  a 
increased 

Beryl 
quartz 

increased 
decreased 

increased 
decreased 

decreased 
decreased 

THERMAL  CHARACTERS. 


171 


GYPSUM  (Monoclinic).  The  angle  2E  decreases  with  rising  temperature  and  the 
principal  vibration  directions  change,  the  dispersion  changing  from  inclined  to  hori- 
zontal. 


Temperature. 

2ZTfor 

Temperature 

2E  for 

Centigrade. 

Na  Yellow. 

Centigrade. 

Li   Red. 

20° 

92° 

20° 

96° 

50° 

79° 

47° 

76° 

100° 

51° 

71° 

59° 

116° 

36° 

95° 

39° 

I349 

0° 

116° 

0° 

The  test  is  usually  made  by  replacing  the  glass  vessel  M,  Fig. 
273,  by  a  metal  air  bath,  Fig.  287,  consisting  of  a  rectangular  hol- 


FJG.  287. 

low  box  of  copper,  which  projects  on  either  side  beyond  the  po- 
lariscope,  and  is  heated  by  gas  burners.  At  the  top  is  an  opening 
for  the  crystal  carrier,  and  at  the  large  sides  are  glass  windows  so 
set  in  tubes,  that  by  a  key  the  distance  apart  can  be  made  as  small 
as  the  crystals  will  permit.  From  the  top  of  the  box  project  two 
thermometers  T reading  to  300°  C. 

The  crystal  section  is  adjusted,  and  the  angle  is  determined'at 
room  temperature,  the  two  burners  are  then  lighted  and  the  heat- 
ing continued  until  a  constant  temperature  can  be  maintained  for 
one-half  hour,  when  a  new  reading  is  made ;  this  is  repeated^  for 
different  intensities  of  flame. 


CHAPTER  X11I. 


THE    MAGNETIC   AND  ELECTRICAL    CHARACTERS 
OF  CRYSTALS. 

MAGNETIC  INDUCTION  OF  CRYSTALS. 

All  substances  are  either  attracted  or  repelled  in  some  degree 
when  in  the  field  of  a  strong  electromagnet.  If  attracted  they  are 
said  to  be  "paramagnetic"  or  "  magnetic \"  if  repelled  they  are 
"  diamagnetic  " 

If  a  rod  of  any  substance  is  suspended  by  a  fibre  so  as  to 
swing  freely  horizontally  between  the  vertical  poles  of  an  elec- 
tromagnet, abt  Fig.  288,  mag- 
netic induction  takes  place  and 
as  the  lines  of  force  between 
the  poles  are  essentially  hori- 
zontal, the  effect  of  the  pull  or 
thrust  upon  rotation  is  greatest 
for  the  particles  furthest  from 
the  axis  of  rotation.  If  para- 
magnetic, therefore,  the  effect 
is  to  pull  the  rod  into  a  longi- 
tudinal or  "axial"  position 
with  its  ends  as  near  the  poles 
of  the  magnet  as  possible,  and 
if  diamagnetic  the  rod  is  pushed 
into  a  transverse  or  "  equator- 
ial" position  with  its  ends  as 
far  from  the  magnetic  poles  as 
possible. 

Fig.  288  shows  the  apparatus 
of  Edmond  Becquerel*  in  which 
A  and  B  are  the  poles  of  a 
large  electro-magnet,  C  and 
C'  square  soft  iron  pole  pieces 
and  DE,  D' E  long  narrow  soft 
FIG.  288.  TV  natural  size.  iron  pole  pieces  placed  so  that 

*  Ann.  de  Chim.  et  de  Phys.y  1850,  V.,  28,  p.  283. 


MAGNETIC  CHARACTERS.  173 

the  back  face  of  D  E  and  front  face  of  Dr  Ef  lie  in  the  same  plane 
through  the  torsion  thread.  Bars  of  25  mm.  long,  2  to$  mm.,  broad 
are  suspended  in  the  position  a  b  and  adjusted  by  the  torsion  head 
N  until  a  scratch  at  one  end  of  the  bar  coincides  with  the  cross 
hair  of  the  telescope  L.  The  current  is  then  turned  on  and  the 
deviation  due  to  attraction  or  repulsion  observed  and  measured  by 
the  amount  it  is  necessary  to  turn  the  circle  H  H'  to  restore  the 
original  position. 

With  crystals,  however,  the  particles  in  certain  directions  be- 
come more  strongly  magnetized  than  in  others,  and  the  para-  or 
diamagnetism  is  judged  by  hanging  a  thin  glass  tube,  filled  with 
powder  of  the  substance,  between  the  magnetic  poles,  the  particles 
of  the  powder  having  all  possible  orientations  all  effect  of  direction 
is  eliminated. 

To  DETERMINE  THE  RELATIVE  STRENGTH  OF  MAGNETIZATION 
IN  DIFFERENT  DIRECTIONS  IN  A  CRYSTAL. 

The  crystal  is  suspended  by  a  silk  fibre  and  should  be  of  such 
a  shape  that  the  section  normal  to  the  axis  of  suspension  is  circu- 
lar. If,  however,  the  crystal  is  small,  the  form  is  less  important. 

Plucker  used  for  his  experiments  a  large  electromagnet  with  six 
Groves  elements,  the  poles  of  the  magnet  being  1.6  inches  apart 
and  the  space  around  the  poles  protected  from  currents  of  air  by  a 
glass  cage.  The  crystal  was  suspended  by  a  double  silk  thread 
from  a  torsion  balance. 

The  direction,  in  the  section  normal  to  the  axis  of  suspension, 
which  is  most  strongly  affected  will  evidently  take  an  axial  posi- 
tion with  paramagnetic  crystals  and  an  equatorial  position  with 
diamagnetic  crystals. 

By  suspending  a  cube  by  its  three  rectangular  axes,  a,  b  and  c, 
successively  the  magnetic  intensities  in  these  directions  may  be 


compared,  for  example: 


Suspended  by : 
a         b         c 


Axial  direction  assumed  by  b         a         b 

Equatorial  direction  assumed  by        c        c         a 

If  the  crystal  is  diamagnetic,  then,  since  c  was  twice  equatorial, 
c  is  the  axis  of  greatest  magnetization,  but  if  paramagnetic,  then  is 
^the  axis  of  greatest  magnetization. 

The  relations  cannot  yet  be  said  to  be  well  understood  as  very 


174  CHARACTERS  OF  CRYSTALS. 

few  minerals  have  been  thoroughly  tested.  It  does  not  appear 
that  isometric  crystals  are  magnetically  isotropic,  for  the  latest  in- 
vestigations of  magnetite  show  that  prisms  cut  parallel  to  a  ternary 
axis  are  most  strongly  magnetized,  those  parallel  to  a  binary  axis 
only  a  little  more  feebly  and  those  parallel  to  a  quaternary  axis 
much  more  feebly.* 

The  experiments  of  Pluckerf  appear  to  prove  that  all  crystals  of 
other  systems  are  magnetically  anisotropic  and  that  a  suspended 
sphere  in  a  uniform  magnetic  field  is  in  stable  equilibrium  only 
when  the  direction  most  strongly  magnetized  is  "  axial,"  that  is,  is 
parallel  to  the  lines  of  force. 

In  hexagonal  and  tetragonal  crystals  the  direction  of  maximum 
magnetization  is  either  parallel  or  at  right  angles  to  the  vertical 
axis  c  and  the  crystal  is  said  to  be  magnetically  positive  or  mag- 
netically negative.  If  a  sphere  is  suspended  with  c  horizontal  then 
four  cases  J  result. 

Paramagnetic  positive,  the  position  of  c  is  axial. 

Paramagnetic  negative,  the  position  of  c  is  equatorial. 

Diamagnetic  positive,  the  position  of  c  is  equatorial. 

Diamagnetic  negative,  the  position  of  c  is  axial. 

In  orthorhombic,  monoclinic  and  triclinic  crystals  the  directions 
of  maximum  and  minimum  intensity,  are  at  right  angles  to  each 
other  and  the  intensity  of  the  third  axis  of  the  "  induction  ellip- 
soid "  is  at  right  angles  to  their  plane. 

In  orthorhombic  crystals  §  these  axes  are  the  crystal  axes  a,  b,  c. 
In  monoclinic  crystal  one  axis  is  parallel  to  bt  the  others  are  in 
oio,  but  not  necessarily  or  probably  parallel  either  to  axes  of 

*  Aimantation  non  isotrope  de  la  Magnetite  cristallisee. — P.  Weiss.  —  C.  R.,  CXXII. 
1405. 

f  Pogg.  Ann.,  v.  72,  p.  315;  v.  76,  p.  576;  v.  77,  p.  447 ;  v.  78,  p.  427 ;  v.  86,  p.  I. 

\  Examples  are : 

-{-Paramagnetic,  — Paramagnetic,  -f- Diamagnetic,  — Diamagnetic, 
Siderite,                  Tourmaline,            Calcite,  Bismuth, 

Wernerite,  Beryl,  Mimetite,  Arsenic, 

Torbernite,  Vesuvianite,  Wulfenite,  Zircon. 

g  Examples  are  : 

Strength  of  Magnetization. 

01  >        02  >  03 

Topaz,  Paramagnetic,  a             c             b 

Anhydrite,  Diamagnetic,  a             b             c 

Barite,  Diamagnetic,  cab 

Epsomite,  Diamagnetic,  c             b             a 


ELECTRICAL  CHARACTERS.  175 

thermal  conductivity  or  optical  principal  vibration  directions.     In 
triclinic  crystals  there  are  no  fixed  relations. 

Practically  no  satisfactory  determinations  have  been    made  of 
absolute  values  of  coefficients. 


TRANSMISSION  OF  ELECTRIC  RAYS.  * 

Electric  waves  differ  from  light  waves  only  in  their  vastly  greater 
length,  they  travel  with  the  same  velocity  and  exhibit  similar  phe- 
nomena. Many  substances  opaque  to  light  waves  are  transparent 
to  electric  waves,  and  in  this  lies  the  hope  of  a  series  of  tests 
for  optically  opaque  crystals  corresponding  to  the  series  for  optic- 
ally transparent  crystals. 

Professor  Bose  |  describes  an  apparatus,  essentially  an  electric 
polariscope,  the  polarizer  and  analyzer  consisting  of  wire  gratings, 
made  by  winding  fine  copper  wire  2  mm.  diameter  around  a  thin 
sheet  of  mica  (about  25  lines  per  cm.) ;  the  mica  pieces  are  then 
dipped  in  melted  paraffine,  after  which  a  round  disc  is  cut  from  the 
sheet.  The  electric  waves  are  produced  by  a  small  Ruhmkorfif 
coil  causing  oscillatory  discharges  between  two  small  (ij^  cm.) 
metallic  spheres ;  beyond  these  in  the  same  tube  is  a  convex  lens 
with  a  spark  gap  at  its  principal  focus,  then  follow  in  order  the 
grating  polarizer,  the  crystal,  the  grating  analyzer,  a  modified 
coherer,  and  a  connected,  distant  d'Arsonval  galvanometer. 

When  the  gratings  are  crossed  no  current  is  shown,  the  in- 
troduction of  the  crystal  produces  a  current  which  is  shown  by 
the  throw  of  the  needle  reflected  by  a  mirror  upon  a  scale.  When 
the  principal  vibration  directions  of  the  crystal  coincide  with  those 
of  the  gratings  no  current  passes.  Crystals  of  moderate  size  are 
successfully  tested  by  this  apparatus. 

ELECTRICAL  CONDUCTIVITY. 

Electrical  conductivity,  although  varying  between  very  wide 
limits  in  different  substances,f  appears  to  be  dependent  much  more 

*  The  Polarization  of  Electric  Rays  by  doubly  refracting  Crystals. — ].  C.  Bose,  Jour. 
Asiatic  Soc.,  Bengal,  LXIV.,  291,  1895. 

f  No  substance  possesses  absolute  electrical  resistance  ;  practically,  however,  con- 
ductivity may  be  considered  to  be  limited  to  the  metals ;  some  metalloids ;  most 
sulphides,  tellurides,  selenides,  bismuthides,  arsenides  and  anlimonides,  some  of  the  ox- 
ides ;  and,  at  higher  temperature,  a  few  haloids. 


176  CHARACTERS  OF  CRYSTALS. 

upon  the  constitution  of  the  chemical  molecule  than  upon  the 
crystalline  structure.  A  certain  dependence  upon  crystallographic 
direction  has,  nevertheless,  been  observed  in  a  few  substances. 

The  principal  experiments  are  those  of  Wortman,*  and  the  more 
recent  series  by  Beijerinck.f  Both  used  essentially  the  same 
method,  in  which  a  prism  of  known  dimensions  was  introduced 
into  a  direct  weak  current,  the  strength  of  which  was  varied  by  re- 
sistances and  the  deviation  observed  in  a  galvanometer. 

Good  contact  was  obtained  by  using  tough  copper  amalgam  or 
sometimes  bright  sheet  lead  or  simply  graphite  rubbed  on  with  a 
lead  pencil.  Natural  faces  were  cleaned  with  acid,  caustic  soda, 
water  and  alcohol,  and  sometimes  even  with  hydrofluoric  acid. 

To  study  the  effect  of  temperature  the  substance  was  heated  in 
a  small  air  bath  made  with  double  walls  of  asbestos  and  covered 
with  copper,  thus  giving  a  very  regular  distribution  of  temperature 
without  thermo-electric  effects. 

The  principal  results  bearing  upon  crystal  structure  are  as  fol- 
lows : 
Isometric  Crystals. 

Magnetite,  the  electrical  conductivity  is  essentially  alike  in  all 
directions. 
Tetragonal  and  Hexagonal  Crystals. 

Hematite,J  hexagonal,  the  electrical  conductivity  parallel  to  c 
is  essentially  twice  that  normal  to  c  and  for  any  direction  making 
the  angle  a  with  c\  the  resistance  is  ^  =  wasinaa  +  wc  cos2a  in  which 
<oa  and  wc  are  the  resistances  normal  and  parallel  to  c. 

Cassiterite,  tetragonal,  and  zincite,  hexagonal,  give  conformable 
results. 
Orthorhombic  Crystals. 

Marcasite  and  bismuthinite,  show  the  greatest  and  least  resis- 
tances parallel  to  two  of  the  axes  ay  b,  c. 

That  is,  the  few  tests  recorded  show  that  the  electrical  conduc- 
tivity of  crystals  conforms  to  the  thermal  conductivity. 

THERMOELECTRIC  CURRENTS. 

If  a  metallic  circuit  is  made  by  soldering  together  one  end  of 
each  of  two  rods  of  different  metals  and  connecting  the  other  ends 

*  Mem.de  la  Soc.  d.  hist.  Nat.  de  Geneve,  XIII.,  1853. 

f  Ueber  das  Leitungsvermogen  der  Mineralien  fur  Eiektricitat. — F.  Beijerinck.— 
Neues  Jahrb.  Min.,  Beil.  Bd.,  XI.,  403,  1896-7. 

{H.  Backstrom.— Ofv.  d.  K.  Vetenskaps-Ak.  Fork,,  1894,  No.  10,  545. 


ELECTRICAL  CHARACTERS.  177 

by  wire,  heating  or  cooling  the  junction  will  develop  an  electric 
current  the  strength  of  which  will  depend  upon  the  change  of 
temperature  and  upon  the  metals  used. 

In  a  crystal  the  electromotive  force  of  the  current  in  part  de- 
pends upon  the  crystallographic  direction,  and  a  thermoelectric 
current  may  be  produced : 

(a)  By  coupling  rods  cut  in  different  directions  from  the  same 
crystal  and  heating  the  junction. 

(b)  By  inserting  a  rectangular  parallelpipedon  longitudinally  in 
a  metallic  circuit  the  sides  being  kept  at  a  constant  equal  tempera- 
ture but  the  ends   differing.      For  example,   Backstrom*  placed 
hematite  between  two  sheet-copper  boxes,  through  one  of  which 
water  was  flowing  and  through  the  other  steam.     The  boxes  were 
connected  by  wires  with  a  Lippman  capillary  electrometer. 

(c)  The  end  surfaces  may  be  held  at  the  same  temperaturef  and 
two  opposite  side  faces  at  different  temperatures. 

Friedel  J  clamped  three  plane  parallel  plates  between  two  plat- 
inum wires,  touching  equal  spaces  on  the  crystal,  connected  the 
plates  with  a  galvanometer  and  placed  them  in  a  space  uniformly 
heated  by  a  water  bath. 

Slight  impurities  appear  to  produce  very  notable  changes  in  the 
thermoelectric  results. 

DIELECTRIC  INDUCTION  IN  CRYSTALS. 

A  crystal  suspended  in  an  electrostatic  field  develops  in  an  al- 
most inconceivably  short  period§  an  electric  polarity,  and  the  crys- 
tal tends  to  assume  a  position  in  which  the  lines  of  force  and  the 
direction  of  maximum  induction  are  parallel. 

According  to  the  experiments  of  Root||  the  directions  of  maxi- 
mum and  minimum  induction  are  respectively  the  principal  vibra- 
tion directions  c  and  a  in  light  transmission.  In  these  experiments 
circular  plates  of  tourmaline,  quartz,  and  calcite,  about  10— II  mm. 
in  diameter  and  cut  parallel  to  the  optic  axis,  were  attached  by  a 
little  drop  of  glue  to  a  silk  thread  and  suspended  between  the 
vertical  plates  of  a  condenser  charged  by  a  rapidly  alternating 

*  Last  cit. 

f  Liebisch,  Grundrissder  Phys.  Kryst.,  217. 
\  Ann.  de  Chim.  et  de  Phys.,  1869,  XVII.,  79. 
$  Less  than  0.0000821  second,  according  to  Root. 
||  Poggendorfs  Annalen,  CLVIII.,  I,  425, 1876. 


CHARACTERS  OF  CRYSTALS. 


current.  The  vibration  through  an  arc  of  10-20  minutes  thereby 
produced  in  the  plate  was  reflected  to  a  telescope  two  meters  dis- 
tant, and  its  period  determined  both  when  the  direction  of  maxi- 
mum induction  was  vertical  and  when  horizontal.  The  quotient 
of  the  former  by  the  latter  gave  a  basis  of  comparison.  Aragonite 
and  topaz  were  also  tested.  For  instance, 


Period  of  vibration  wher 
indue 
A,  Vertical. 

i  direction  ot   maximum 
tion  is 
B.  Horizontal. 

Quotient  of  4 
/> 

Aragonite 
« 

Calcite 

4.1125 
4.0500 
2.240 

3-768 
3.668 

2.175 

I.OpI 
1.105 
1.022 

The  condenser  Fig.  289  consisted  of  a  ring  C  of  gutta  percha 
serving  as  a  frame  for  the  two  vertical  brass  plates  A  and  B. 
Through  C  passed  the  glass  tube  D  with  a  torsion  head  to  which 

the  discs  were  hung  between  A 
and  B  by  a  single  thread  of  silk. 
Tile  motions  of  the  disc  were 
observed  through  the  opposite 
glass  windows  E.  A  voltaic  cur- 
rent passing  through  a  commu- 
tator produced  alternately  -f  and 
—  charges  in  A  and  B  in  some 
instances  as  rapidly  as  6,000  al- 
ternations per  second. 

As  all  dielectrics  have  some 
degree  of  conductivity  the  elec- 
tric polarity  is  apt  to  be  modified 


FIG.  289. 


by  the  more  slowly  developed 
conductivity  phenomena. 


The  effect  of  conductivity  in  use  of  the  constant  current  was 
shown  by  suspending  a  calcite  plate  with  c  horizontal,  c  was  at 
first  axial  but  in  less  than  two  minutes  turned  reversing  the  poles, 
whereas  with  alternating  current  c  was  equatorial. 

The  strength  of  the  induction  in  any  direction  is  indicated  by 
the  so-called  dielectric  constant*  which  may  be  determined  by 

*  Maxwell  assumes  to  be  proportionate  to  the  square  of  the  constant  A  in  Cauchy's  for- 

n 

mula  for  dispersion,  n  =  A-\ or,  since  if  A  =  oo  n  =  A,  to  the  square  of  the  index  of 

refraction   for  light  of  infinite   wave-length   vibrating  parallel   the   given  direction. 
(Groth,  Phys.  Kryst.,  III.  Ed.,  194.) 


ELECTRICAL  CHARACTERS.  179 

the  method  of  Bolzmann  *  in  which  the  attractions  exerted  are 
measured  as  follows  :  A  small  sphere  of  the  dielectric  to  be  tested 
is  attracted  by  a  fixed  charged  metal  sphere  and  the  amount  of  its 
deviation  h  determined  by  the  deflection  of  a  mirror;  the  sphere  is 
then  replaced  by  another  sphere  of  the  same  diameter  but  coated 
with  tin  foil  and  the  deviation  h'  of  this  produced  by  the  same 
charge  in  the  metal  sphere  is  determined  ;  then,  according  to  Bolz- 
mann, if  s  is  the  dielectric  constant 

h' 


either  for  uniaxial  crystals  or  for  biaxial  crystals  in  which  an  axis 
of  dielectric  symmetry  is  parallel  to  the  lines  of  force. 

The  constants  may  also  be  determined  by  comparison  ot 
capacity.f 

From  the  dielectric  constants  in  the  principal  directions  the  con- 
stants for  any  direction  may  be  calculated.^ 

If  DaDbDc  are  the  principal  constants  the  constant  Dr  for  any  di- 
rection is  : 

Da  cos2  (a.f)  +  D  cos2  (b  r)  -*-  Dc  cos2  (c-r) 

The  conductivity  on  the  plane  surface  of  a  dielectric  crystal  theoretically  must  con- 
form to  the  dielectric  constants  in  different  directions. 

Wiedemann  's$  Experiment.  —  An  insulated  needle  was  fastened  in  proper  holder  in 
normal  contact  to  a  crystal  face,  previously  covered  with  a  poorly  conducting  powder 
such  as  lycopodium  or  minium,  and  positively  charged  by  contact  with  the  knob  of 
a  leyden  jar.  The  powder  near  the  point  is  tossed  aside  in  shapes  which  are  elliptical 
or  circular  according  to  surface.  The  results  can  be  made  permanent  by  collodion.  In 
general  the  longest  axis  is  in  the  vibration  direction  of  the  light  of  greatest  velocity. 

de  Senarmonf  s\\  Experiments.  —  de  Senarmont  coated  the  crystal  with  tin  foil  except 

that  a  circular  hole  was  cut  in  the  foil  covering  the  face  to  be  studied.     The  foil  was 

grounded  the  crystal  was  placed  in  partial  vacuum  opposite  a  point  of  brass  wire  from 

which  positive  electricity  was  discharged  following  a  direction  assumed  therefore  to  be 

»  that  of  easiest  conductivity.     The  tests  are  made  in  the  dark. 

*  Ber.  Ak.  Wien.  (2),  LXX.,  342,  1874,  also  Ch.  Borel,  Arch.  Soc.  Fhys.  et  nat.  de 
Geneve  (3),  XXX.,  131,  1893. 

f  Liebisch,  Grundriss  der  Phys.  Kryst.,  230. 

J  A.  Lampa,  Ber.  Ak.   Wein.,  CIV.,  1179. 

\p°gg.  Ann.,  76,  404,  1849. 

||  de  Senarmont,  Ann.  de  Chim.  et  de  Phys.  (3),  28,  257,  1850. 


i8o  CHARACTERS  OF  CRYSTALS. 

PYRO-ELECTRICITY. 

Equal  positive  and  negative  charges  of  electricity  are  developed 
at  different  points  'or  poles  of  certain  crystals  during  a  uniform 
change  of  temperature. 

A  temperature  change  of  at  least  70°  to  80°  C.  is  desirable. 
Usually  the  crystal  is  heated  in  an  air  bath  to  a  uniform  tempera- 
ture, then  drawn  quickly  once  or  twice  through  an  alcohol  flame 
to  remove  any  electricity  occurring  on  the  surface,  and  then  brought 
into  a  cooler  place  and  allowed  to  cool. 

If  heating  injures  the  crystal  it  rr.ay  be  cooled  from  the  room 
temperature  by  a  freezing  mixture.* 

During  the  cooling  of  the  crystal  the  positive  charges  collect  at 
the  so-called  antilogue  poles,  and  the  negative  charges  at  the  ana- 
logue poles,  f  and  may  be  distinguished  by  their  effect  on  other 
electrified  bodies.  For  instance,  a  cat's  hair  rubbed  between  the 
fingers  becomes  positively  electrified  and  is  attracted  by  the  analo- 
gous pole  and  repelled  by  the  antilogous  pole,  or  as  in  the  method 
of  Hankel,J  the  poles  may  be  touched  by  a  platinum  wire  care- 
fully insulated  and  worked  by  a  system  of  levers,  and  the  charge 
conducted  to  an  especially  constructed  gold  leaf  electrometer. 

Kundt's§  method  is,  however,  most  generally  employed  and  con- 
sists in  blowing  upon  the  cooling  crystal  a  fine  well  dried  ||  mix- 
ture of  equal  parts  of  powdered  sulphur  and  red  oxide  of  lead. 
The  nozzle  of  the  bellows  is  covered  by  a  muslin  net  and,  in 
passing  through,  the  sulphur  is  negatively  electrified  and  is  attracted 
by  the  positive  poles  coloring  them  yellow,  while  the  minium  is 
positively  electrified  and  is  caught  by  the  negative  poles  coloring 
them  red.  By  pressing  the  crystal  upon  sticky  paper  a  permanent 
record  can  be  obtained.  The  dust  should  fall  evenly  and  the  bel- 
lows be  held  far  enough  away  to  prevent  direct  action  of  the  blast. 

The  figures  here  shown^[  represent  crystals  dusted  with  sulphur 
and  minium  during  cooling,  the  darker  dotted  portions  showing 
the  analogue  poles,  the  hatched  portions,  the  antilogue  poles. 

*  Snow  or  ground  ice  and  salt ;  3  pts.  snow  or  ground  ice  with  I  pt.  H2SO4 ;  2  ptsr 
snow  or  ground  ice  with  3  pts.  crystals  CaCl2. 

\  With  rising  temperature  these  are  reversed. 

\  G.  W.  Hankel,  Inaug.  Dissertation. 

\  Fogg.  Ann.,  CXXXVI.,  612,  1862;  Weid.  Ann.,  XX.,  592,  1883;  XXV.,  145. 
1886. 

||  Dry  over  H2SO4  in  a  vessel  from  which  the  air  has  been  partially  exhausted. 

\  Drawn  from  the  colored  plate  III.  in  Groth,  Phys.  Kryst.,  III.  ed. 


ELECTRICAL  CHARACTERS. 


181 


Fig.  290  represents  a  tourmaline  crystal,  Fig.  291  a  calamine 
crystal,  in  both  of  which  polarity  is  shown  with  reference  to  the 


FIG.  290. 


FIG.  292. 


vertical  axis.  Fig.  292  represents  a  boracite  crystal,  showing 
polarity  with  reference  to  four  axes,  and  Fig.  293  a  quartz  crystal 
with  three  axes  of  polarity.  Fig.  294  shows  a  basal  section  of 
quartz. 


FIG.  293. 


FIG.  294. 


FIG.  295. 


Quantitative  examinations  were  made  by  Gaugain*  by  coating 
the  ends  of  a  cylinder  of  the  crystal  with  tin  foil,  connecting  one 
end  to  the  ground  and  the  other  to  some  form  of  self-discharging 
electroscope.  The  number  of  discharges  are  counted  and  serve 
as  a  relative  measure  especially  if  the  capacity  of  the  electroscope 
is  small.  In  this  way  it  was  shown  that  with  tourmaline  the 
amount  is  independent  of  the  time  of  cooling,  is  alike  for  the 
same  change  of  temperature  in  either  direction,  and  is  independ- 
ent of  the  length,  but  proportionate  to  the  cross  section  and  to  the 
difference  between  the  end  temperatures. 

THEORY  OF  LORD  KELVIN. 

Lord  Kelvin  theorized  f  that  tourmaline,  in  which  pyroelec- 
tricity  was  first  observed,  is  internally  in  a  state  of  uniform  electric 
polarity,  and  that,  therefore,  at  the  surface  there  should  be  an 

*  Ann.  de  Chim.  et  de  Phys.,  III.,  57. 

fMath.  phys.  papers  Sir.  William  Thomson,  I.,  315. 


1 82  CHARACTERS  OF  CRYS1ALS. 

electric  charge  of  uniform  surface  density.  In  a  medium  not 
perfectly  insulating  such  as  damp  air  there  is  gradually  formed  by 
induction  an  electric  layer  completely  neutralizing  the  charge  in 
the  tourmaline,  that  is  making  it  essentially  non-electric. 

But  if  the  tourmaline  is  heated  or  cooled  the  strength  of  the 
internal  polarization  is  altered,  while  that  of  the  electrified  layer  of 
outer  medium  is  more  slowly  altered  and  the  effect  is  electric  polarity. 

Riecke  *  confirmed  this  theory  by  showing  that,  in  a  fairly  per- 
fectly insulating  medium,  tourmaline  remained  electrified  twenty 
to  thirty  hours,  after  cooling  to  its  normal  temperature.  The  heated 
tourmaline  while  still  non-electrified  was  placed  under  an  air  pump 
over  a  gold  leaf  electroscope  and  the  air  slightly  rarefied. 

PIEZO-ELECTRICITY. 

Electric  charges  may  be  developed  by  pressure,  for  instance, 
calcite  pressed  between  the  fingers  becomes  positively  electrified, 
tourmaline  compressed  in  the  direction  of  c  shows  a  positive 
charge  at  the  antilogue  end  and  a  negative  charge  at  the  analogue 
end  or  precisely  the  charge  which  would  result  from  cooling ; 
whereas,  if  the  pressure  is  removed  and  the  crystal  allowed  to 
expand  the  charges  are  reversed  conforming  to  rising  temperature. 
The  quantity  developed  is  always  proportionate  to  the  pressure. 

If  strains  are  developed  by  unequal  heating  electrical  charges 
may  be  developed  as,  for  instance,  by  standing  a  basal  section  of 
quartz  upon  a  hot  centrally  placed  metal  cylinder,  the  distribution 
of  charges  conforming  to  the  effect  of  cooling  a  complete  crystal, 
whereas,  with  a  hot  metal  ring  the  charges  are  reversed. 

The  methods  for  detecting  the  charges  are  the  same  as  for 
pyroelectricity.  Fig.  295  shows  the  distribution  of  the  charges  in 
a  basal  section  of  quartz  produced  by  pressure  in  the  direction  of 
the  arrows  which  conforms  evidently  to  Fig.  294,  or  the  effect  of 
cooling. 

For  quantitative  examination  J.  and  P.  Curie  used  the  following 
method.t 

The  crystal  is  cut  in  the  form  of  a  rectangular  parallelopipedon, 
Fig.  296,  and  two  opposite  faces,  A,  B,  covered  with  tin  foil.  One 
of  these,  A,  is  grounded ;  the  other,  B,  is  connected  with  one  of 
the  plates  of  a  condenser  C  of  known  capacity,  and  also  with  one 

*  E.  Riecke,  Wied,  Ann.  28,  43, 31,  889,  40,  264,  ^9,421. 
fCompte  Rendu,  XCL,  294,  383 ;  XCII.,  350. 


ELECTRICAL  CHARACTERS. 


183 


of  the  couples  55  of  a  Thomson-Mascart  electrometer.     The  other 

plate  of  the  condenser  is  grounded,  and  the  couple  5' 5'  of  the 

electrometer    is    connected 

with  one  pole  of  a  Daniell 

cell,  the  other  pole   being 

grounded. 

The  relation  between  the 
pressure  employed  and  the 
electricity  developed  is  de- 
termined as  follows :  Let 
D  denote  the  potential  of 
the  Daniell  cell,  C  the  ca- 
pacity of  the  condenser,  and 
.c  the  capacity  of  the  system, 


FIG.  296. 


consisting  of  the  plate  B,  the  couple  55  and  the  conductors. 

For  some  pressure  Pt  in  the  direction  of  the  arrows,  the  needle 
of  the  electrometer  will  be  at  zero  and  the  entire  system  charged 
to  a  potential  D.  That  is,  the  pressure  will  have  developed*  a 
quantity  Q=(C+c)£>. 

If  the  condenser  is  removed  the  pressure  necessary  to  bring  the 
needle  again  to  the  zero  position  will  be  Pt  and  the  quantity  de- 
veloped will  be  Q'  =  cD,  hence  for  the  pressure  P—  F  the 
quantity  developed  is  Q  —  Q'  =  CD,  that  is  a  quantity  sufficient  to 
charge  a  condenser  of  known  capacity  C  to  a  known  potential  D. 

THEORY  OF  PYRO-  AND  PIEZO-ELECTRICITY.! 

Whenever  the  volume  of  a  crystal  is  altered  either  by  a  tempera- 
ture change  or  by  mechanical  pressure  a  portion  of  the  heat 
energy  or  mechanical  energy  may  be  converted  into  electric  en- 
ergy, which,  in  poorly  conducting  crystals,  will  frequently  be  mani- 
fested by  the  accumulation  of  positive  and  negative  charges  at 
different  points  or  poles. 

Conversely,  as  shown  by  Lippmann,J  a  section  which  becomes 
for  instance,  positively  electric  by  normal  pressure  must,  if  charged 
with  positive  electricity,  expand  in  the  direction  of  the  normal.  In 
other  words,  the  electric  charges  are  functions  of  the  change  in 
^volume. 

*  Quantity  equals  capacity  times  potential. 
fW.  Voigt,  Wicd.  Ann.,UV.,  701,  1895. 
J  G.  Lippman,  Ann.  (.him.  et  Phys,,  (5)  XXIV,  145. 


1 84  CHARACTERS  OF  CRYSTALS. 

J.  and  P.  Curie  devised*  a  delicate  apparatus  for  measuring  the 
force  of  such  expansions  and  the  measured  results  for  quartz, 
checked  almost  absolutely  with  the  calculated  results. 

Crystals  symmetrical  to  a  central  point  (Groups  2,  5,  8,  13,  15, 
21,  22,  25,  27,  29,  30,  32)  cannot  develop  opposite  charges  at  the 
ends  of  a  diameter.  The  apparent  exceptions  obtained  by  Harikel 
and  others  have  not  all  been  explained,  but  may  in  part  be  due  to 
influences  dependent  on  the  method  of  conducting  the  operation,  in 
other  cases,  barite,  for  instance,  they  have  been  ascribedf  to  twin 
structure. 

EFFECT  OF  ELECTROSTATIC  FIELD  UPON  OPTICAL  CHARACTERS. 

Kundt  J  coated  two  of  the  sides,  parallel  c,  of  a  quartz  parallel- 
opiped,  with  tin  foil,  connected  these  with  the  electrodes  of  a  Holz 
machine  and  found  that  the  circular  interference  rings  became 
elliptical,  the  minor  axes  of  the  ellipses  being  parallel  to  the  direc- 
tion of  expansion  produced  by  the  charge.  Pockels  §  shows  that 
the  change  is  too  great  to  be  simply  a  consequence  of  the  expan- 
sion and  must  be  in  part  due  to  a  direct  influence  of  the  electric 
force  on  the  light  motions. 

*  Brief  description  Liebisch,  Grundriss  der  Phys.  Kryst.,  481. 

t  Beckenkamp,  Zeit.  f.  Kryst.,  XXVII.,  85. 

%  Wied.  Ann.,  XVIII.,  228,  1893. 

\  F.  Pockels,  Abh.  d.  k.  Ges.  a.  Gottingen,  XXXIX.,  1-204. 


CHAPTER  XIV. 

ELASTIC  AND  PERMANENT  DEFORMATION  OF 
CRYSTALS. 

In  an  elastic  substance  the  distance  apart  and  relative  position 
of  the  particles  may  be  changed  by  mechanical  force,  and  up  to  a 
certain  so-called  "  elastic  limit "  the  particles  will,  on  removal  of 
the  force,  regain  their  former  position.  Such  a  change  is  called  an 
elastic  deformation  or  form-alteration. 

Any  strain  or  pressure  in  excess  of  the  so-called  "  elastic  limit  " 
produces  a  permanent  change  of  form. 

HOMOGENEOUS    ELASTIC  DEFORMATION. 

The  only  recorded  experiments  by  pressure  on  all  sides  were  made 
upon  halite  by  aid  of  a  piezometer.*  The  cubic  compression 
coefficients  have,  however,  been  calculated  by  Voigt  for  a  number 
of  species.  The  same  five  classes  result  as  with  the  homogeneous 
deformation  produced  by  changes  of  temperature. 

The  alteration  produced  by  a  change  of  temperature  cannot  be 
exactly  compensated  by  uniform  pressure  on  all  sides  except  in  the 
case  of  isometric  crystals. f 

ELASTIC    DEFORMATION  DUE    TO  PRESSURE    IN    ONE 
DIRECTION. 

Observation  shows  that  the  extension  or  compression  of  any  rod 
of  length  /,  breadth  b  and  thickness  /,  produced  by  a  weight  W 

WL 

acting  in  the  direction  of  the  length,  is  -rr  E,  in  which  E  is  a  char- 
acteristic factor  called  the  coefficient  of  extension.  Obviously  if 
W,  /,  b  and  /  are  unity,  the  extension  becomes  equal  E,  that  is  the 
coefficient  of  extension  is  the  extension  produced  upon  a  unit  rod 
by  a  unit  weight. 

*R6ntgenand  Schneider,  Wied.  Ann.,  XXXI.,  1000,  1887. 
f  Liebisch,  Phys.  Kryst,     1891,  576. 


1  86  CHARACTERS  OF  CRYSTALS. 

The  most  convenient  method  of  obtaining  the  extension  coeffi- 
cient for  any  direction  in  a  crystal  is  by  cutting  a  thin  rod  R,  Fig, 
297,  of  rectangular  cross  section,  in  the  given  direction,  supporting 

it,  as  shown,  upon  the  two  rests 
A  and  B,  and  loading  by  a  weight 
W  connected  with  a  centrally 
placed  knife  edge.  The  deflec- 
tion of  the  central  section  pro- 
duced by  the  weight  is  deter- 
mined by  Koch  as  follows:*  A 

FIG  2  rectangular    glass   prism    P    is 

placed  just  below  and  with  one 

face  parallel  to  the  rod,  and  another  face  vertical.  Monochro- 
matic light  is  reflected  upon  the  hypothenuse  face,  and,  very  much 
as  in  the  Fizeau  apparatus,  p.  167,  interference  bands  are  pro- 
duced by  the  rays  reflected  from  the  lower  surface  of  the  rod  R 
and  the  upper  surface  of  the  prism  P,  and  are  viewed  by  a  horizon- 
tal telescope  through  the  vertical  face. 

As  the  rod  is  bent  an  interference  band  coincides  with  the  cross 
hair  whenever  the  thickness  of  the  air  film  is  an  odd  multiple  of  J^A 
for  the  light  used,  that  is  the  depression  corresponding  to  the  inter- 
val between  two  successive  bands  coincident  with  a  cross  hair  is  for 
sodium  light  589.5  millionths  of  a  millimeter. 

Denoting  the  central  depression  produced  by  any  weight  W 
by  n  the  coefficient  of  extension  in  the  direction  of  the  length  of 
the  rod  is  given  by  the  formulaf 


SURFACE  OF  EXTENSION  COEFFICIENTS.     Nine  classes  result  from 
the  measurements  of  extension  coefficients  of  crystals. 

1.  Isometric  crystals  all  yield  an  extension  surface  symmetrical  to- 
nine  planes  and  thirteen  axes.     The  four  central  sections  parallel 
to  the  octahedral  planes  are  circles.     The  cubic  axes  are  direc- 
tions of  maximum  or  minimum  extension  and  the  octahedral  axes 
are  correspondingly  minimum  or  maximum. 

2.  Hexagonal.     Classes  19,  22,  23,  24,  25,  26,  27.     The  exten- 
sion surface  is  a  surface  of  rotation  around  c  .     Each  vertical  cen- 
tral plane  and  the  horizontal  central  plane  are  planes  of  symmetry, 

*  Wied.  Ann.,  XVIIL,  325,  1883. 
f  Groth,  Phys.  Kryst.,  III.    ed.,  203. 


ELASTIC  DEFORMATION. 


187 


3.  Hexagonal.     Classes  18,  20,  21.     The  extension  surface  has 
one  circular  section  horizontally  through  the  center.     In  general 
the  shape  is   that   of  a   rhombohedron 

with  rounded  edges  and  angles  symme- 
trical to  three  vertical  planes  and  to  a 
ternary  vertical  axis.  Fig.  298  shows 
the  section  of  the  surface  for  calcite  made 
by  a  principal  section  normal  to  a  rhombic 
face,  the  dotted  lines  being  the  directions 
of  greatest  and  least  extension.  Fig.  299  shows  the  section  nor- 
mal to  this. 

4.  Hexagonal.     Classes  1 6  and  17.     Like  No.  3  but  without  the 
three  vertical  planes  of  symmetry. 

5.  Tetragonal.     Classes  n,  12,  14  and  15. 

6.  Tetragonal.     Classes  9,  10  and  13. 

7.  Orthorhombic.     The  extension  surface  symmetrical  to  the  three 


FIG.  298.         FIG.  299. 


FIG.  300. 


FIG.  301. 


FIG.  302 


pinacoidal  planes.     Figs.  300,  301,  302,  show  these  sections  for 
barite. 

8.  Monoclinic.     The  extension  surface  symmetrical  to  the  clino- 
pinacoid  and  to  the  axis  b. 

9.  Triclinic.     No  investigations  are  as  yet  recorded. 

EFFECT  OF  PRESSURE  IN  ONE  DIRECTION  UPON  THE  OPTICAL  CHAR- 
ACTERS. 

If  a  rectangular  parallelopipedon*  C,  Fig.  303,  is  compressed  be- 
tween the  parallel  jaws  of  a  screw  press  f  in  general  the  velocity  of  the 

ight  vlibrating  parallel  to  the 
pressure  is  increased.  If  the  pres- 
sure is  uniform  the  convergent 
light  effects  are  studied  by  bring- 
ing the  specimen  over  the  stage 
of  a  polariscope,  but  if  at  all  un- 


FIG.  303. 


*  Pockels  used  rectangular  parallelepipeds  13  mm.  high,  25.5  broad  and  thick  and 
compressed  from  30  k.  g.  with  quartz  to  2  k.  g.  for  sylvite  per  sq.  mm. 

fFor  press  recording  pressure  exerted  see  Groth,  Phys.  Kryst.,  III.  ed.,  215. 


1 88  CHARACTERS  OF  CRYS1ALS. 

equal,  determinations  in  parallel  light  of  vibration  directions,  faster 
and  slower  ray  and  strength  of  double  refraction  are  more  safe. 

Amorphous  Substances. 

Glass*  gives  under  compression  a  negatively  uniaxial  figure  the 
optic  axis  parallel  to  the  direction  of  pressure ;  that  is  c  parallel  a, 
or  the  ray  vibrating  parallel  to  the  pressure  is  the  more  rapidly 
transmitted. 

Optically  Isotropic  Crystals. 

In  general  become  biaxial  but  when  the  pressure  is  applied 
parallel  to  the  cubic  (quaternary)  or  octahedral  (ternary)  axes  the 
crystal  becomes  uniaxial. 

Optically  Uniaxial  Crystals  (Denoting  by  a,  b,  c  the  principal  vibra- 
tion directions  and  also  velocities). 

In  positive  crystals  c  is  parallel  to  c  and  a  =  b  >  c,  while  in  nega- 
tive crystals  c  is  parallel  to  a  and  a  >  b  =  c-  Pressure  parallel  to 
c,  in  positive  crystals,  will  cause  c  to  approach  a  and  possibly  to 
equal  (isotropy)  or  exceed  it  (negative) ;  while  in  negative  crys- 
tals pressure  will  only  make  a  still  greater.  In  either  positive  or 
negative  crystals  pressure!,  parallel  a,  will  make  the  three  veloci- 
ties a,  b,  c  unequal,  that  is  develope  a  biaxial  structure. 

Optically  Biaxial  Crystals. 

The  relations  are  more  complex.  Pressure  perpendicular  to 
the  plane  of  the  optic  axis  decreases  the  axial  angle  in  positive 
crystals,  and  the  crystal  may  be,  as  with  heat,  p.  170,  for  a  certain 
pressure  uniaxial  or  the  axes  may  pass  into  a  plane  at  right  angles 
to  their  former  position ;  with  negative  crystals  the  axial  angle  is. 
increased.  Pressure  parallel  to  the  obtuse  bisectrix  will  in  positive 
crystals  increase  the  axial  angle  and  in  negative  crystals  decrease  it. 

CLEAVAGE. 

In  crystals  the  elastic  limit  varies  with  the  direction,  that  is,  in 
certain  directions  there  is  a  weaker  cohesion  of  the  particles  than 
there  is  in  others,  and  many  crystals  tend  to  separate  or  cleave 

*  If  the  pressure  is  not  uniformly  applied  at  all  points  of  the  opposite  surface  the 
effect  may  resemble  biaxial  lemniscates. 

f  The  centre  of  each  ring  system  thus  developed  in  quartz  is  colored,  that  is  parallel 
to  each  optic  axis  there  is  a  rotation,  and  it  has  been  shown  that  in  these  directions 
t  wo  elliptically  polarized  waves  are  transmitted. 


PERMANENT.  DEFORMATION. 


189 


along  more  or  less  smooth  plane  surfaces  normal  to  these  direc- 
tions of  weaker  cohesion.  This  is  undoubtedly  due  to  the  fact 
that  whatever  form  of  reg- 
ular  grouping  of  particles 
may  exist  in  a  crystal,  ad- 
jacent molecular  planes  in 
certain  directions  will  be 
further  apart  than  adjacent 
planes  in  other  directions, 
and,  therefore,  held  together 
with  less  force.  For  in- 
stance, in  Fig.  304,  it  is  evi- 
dent that  the  closer  the  par- 


FIG.  304. 


tides  are  along  any  direction  abt  cd,  efy  gh  the  greater  is  the  distance 
to  the  adjacent  row  a'&',  c'd',  etft,gthl ,  and  as  undoubtedly  the  fur- 
ther apart  the  particles  are  the  weaker  their  cohesion,  cleavage 
will  be  most  likely  to  occur  parallel  to  the  planes  in  which  the 
crystal  molcules  are  most  closely*  packed. 

The  same  general  relations  exist  for  elastic  and  permanent  de- 
formation.t  Cleavage  will  therefore  be  expected  normal  to  the  di- 
rection of  greatest  elastic  extension,  therefore  parallel  to  faces  with 
simple  indices.  The  possible  cleavage  directions  are : 


System.  Cleavage  Form. 

i    Isometric.  Hexahedron. 

Octahedron. 

Rhombic  Dodecahedron. 
Hexagonal.  Basal  Pinacoid. 

Hexagonal  Prism. 
Rhombohedron. 
Hexagonal  Pyramid  (rarely). 
Tetragonal.  Basal  Pinacoid. 

Tetragonal  Prism. 
Tetragonal  Pyramid  (rarely). 
Orthorhombic.        Pinacoid. 

Prisms  or  Domes. 
Pyramid. 

Monoclinic.  Clinopinacoid. 

Basal  pinacoid. 
Orthopinacoid. 
Orthodome. 
Prism. 

Pyramid  (rarely). 

In  triclinic  crystals  there  can  only  be  equally  easy  cleavage  parallel  to  one  plane. 
Nevertheless  if  there  is  a  direction  of  nearly  perfect  cleavage  one  of  the  principal  vi- 
bration directions  will  be  approximately  normal  to  this  and  the  two  others  parallel 
thereto. 

*  The  densest  planes  are  usually  the  faces  of  common  forms  with  simple  indices, 
f  Groth,  Phys.  Kryst.,  III.  ed.,  229. 


Examples. 
Galenite. 

Fluorite,  diamond. 
Sphalerite. 
Beryl,  pyrosmalite. 
Nephelite,  apatite. 
Calcite,  siderite. 
Pyromorphite. 
Apophyllite. 
Rutile,  wernerite. 
Scheelite. 
Anhydrite,  topaz. 
Barite. 
Sulphur. 

Orthoclase,  gypsum. 
Muscovite,  orthoclase. 
Epidote. 
Epidote. 

Pyroxene,  amphibole. 
Gypsum. 


CHARACTERS  OF  CRYSTALS. 


GLIDING  PLANES. 

In  those  directions  ef,gh,  Fig.  304  in  which  the  adjacent  molecular 
planes  are  closest  together  and  therefore  most  strongly  held  to- 
gether it  sometimes  happens  that  under  tangential  pressure  the 
particles  glide  or  rotate,  without  separation,  into  a  new  position  of 
equilibrium. 

This  phenomenon  has  been  observed  in  calcite,  pyroxene,  anhy- 
drite, stibnite  and  other  minerals. 

In  calcite  for  instance  let  ab'c'd  Fig.  305  represent  a  section 
through  the  optic  axis  normal  to  a  rhombic  face  (see  abed,  Fig. 
222)  then  will  acft  the  optic  axis,  make  an  angle  of  63°  45'  with  adt 
the  direction  of  — y2  R,  and  an  angle  of  45°  23^'  with  ab1  the 
short  diagonal  of  a  cleavage  face. 


FIG.  305.  FIG.  306. 

If  pressure  is  applied  gradually  in  the  direction  of  the  arrows, 
that  is  parallel  to  adt  there  will  be  produced,  about  some  plane  ef 
parallel  to  ad,  a  gliding  or  rotation  of  the  particles  until  eV  has  taken 
the  position  eb  at  which  feb  =  180°  —feb'  =  70°  5 1  y2! 

In  this  rotated  portion  the  optic  axis  will  be  bdr  making  with 
the  optic  axis  ac'  of  the  unchanged  portion  an  angle  of  52°  30'  as 
shown. 

Instead  of  rotation  about  a  single  plane  ef  there  may  be  rotation 
about  several  parallel  planes  as  in  Fig.  306  producing  twin  lamellae. 

If  a  little  calcite  cleavage  is  placed,  as  in  Fig.  307,  with  an  edge 
ad  of  a  larger  angle  resting  upon  a  steady  support  and  the  blade  of 
a  knife  is  pressed  steadily  down  at  some  point  i  of  the  opposite 


PERMANENT  DEFORMATION. 


191 


edge  the  portion  of  the  crystal  between  z  and  c  will  be  slowly 
pushed  as  indicated  into  a  new  position  of  equilibrium  as  if  by  rota- 
tion about  fghm,  or  — y2R,  until  the  new  face^vr'/z  and  the  old  face 
gch  make  equal  angles  with  fghm.  If  carefully  done  gcfh  will  be  a 
perfect  plane ,  but  more  frequently  it  is  step  like  and  the  rotated 
portion  is  apt  to  separate  at  the'gliding  plane. 


FIG.  307.  FIG.  308. 

In  the  described  phenomenon  of  gliding  the  particles  evidently 
assume  new  relative  positions.  If  however  a  rod  of  ice  of  square 
cross  section  is  cut  with  one  pair  of  long  faces  parallel  to  the  optic 
axis  the  other  perpendicular  to  the  optic  axis  and  a  bending  pres- 
sure, Fig.  308,  applied  normal  to  the  axis  there  is  no  change  in  the 
direction  of  the  optic  axis,  indicated  by  the  arrows,  nor  any  especial 
point  at  which  the  movement  ceases ;  that  is  the  particles  have  evi- 
dently slipped*  without  change  in  orientation.  This  has  been 
called  TRANSLATION. 

PARTING. 

The  planes  along  which  a  slipping  has  occurred,  although  pre- 
viously directions  of  maximum  normal  cohesion  may  thereafter  be 
planes  of  easy  separation  or  Parting,\  differing 
from  true  cleavage,  however,  because  in  parting 
the  easy  separation  is  limited  to  the  planes  of 
actual  molecular  disturbance  while  true  cleavage 
is  obtained  with  equal  ease  in  any  part  of  the 
crystal. 
PERCUSSION  FIGURES. 

If  a  rod  with  a  slightly  rounded  point  is  pressed          FIG.  309. 

*Mugge,  Neues  Jahrb.f.  Min.,  1895,  JI-»  2I1- 

f  Similar  planes  of  easy  separation  may  be  due  to  other  causes,  for  instance, 
during  the  growth  of  a  crystal  the  planes  at  certain  intervals  may  be  coated  with  dust 
or  fine  lamellae  of  a  foreign  substance  and  later  the  crystal  may  grow  further.  This 
may  be  repeated  several  times  forming  thus  parallel  planes  of  easy  separation,  e.  g.t 
capped  quartz. 


192  CHARACTERS  OF  CRYSTALS. 

against  a  firmly  supported  plate  of  mica  and  sharply  tapped  with 
a  light  hammer,  three  planes  of  easy  separation  will  be  developed 
as  indicated  by  little  cracks  radiating  *  from  the  point,  Fig.  309. 
One  of  these  is  always  parallel  to  the  ortho  axis  b,  the  others  are 
at  a  definite  angle  x  thereto  of  53°  to  56°  in  muscovite,  59°  in 
lepidolite,  60°  in  biotite,  61°  to  63°  in  phlogopite. 

In  halite,  in  the  same  way,  on  cube  faces  a  cross  is  developed 
with  the  arms  parallel  to  the  diagonals  of  the  face,  while  on  an 
octahedral  face  a  three-rayed  star  is  developed. 

CORROSION  AND  ETCHING. 

Liquids  or  gases  do  not  dissolve  or  attack  chemically  a  crystal 
with  equal  rapidity  in  all  directions.*]" 

This  may  be  shown  experimentally  by  treating  a  sphere  of  the 
crystal  in  a  solvent  as,  for  instance,  a  sphere  of  quartz  in  hydro- 
fluoric acid ;  or  plates  of  different  orientation  may  be  placed  for 
an  equal  time  in  the  same  solvent  and  their  decrease  in  thickness 
compared. 

ETCH  FIGURES. 

The  attack  of  any  liquid  or  gas  upon  any  crystal  face  does  not 
commence  at  the  same  time  at  all  points  but  proceeds  from  certain 
points  first  and  later  from  others. 

From  each  point  of  attack  the  action  proceeds  with  different 
velocities  in  crystallographically  different  directions  and  if  stopped 
at  the  right  time  the  face  will  be  found  to  be  pitted  with  little  an- 
gular etch  cavities  of  definite  shape.  As  these  increase  in  number 
and  size  they  finally  reach  a  stage  when  they  either  touch  or  are 
separated  only  by  little  elevations,  etch  hills,  the  sides  of  which  are 
the  faces  of  the  etch  cavities. 

SHAPE  OF  ETCH  FIGURES. 

Provided  the  same  conditions  exist  of  solvent,  time  and  tem- 
perature, the  etch  figures  developed  on  any  one  face  of  a  crystal 
will  be  of  the  same  shape  and  in  parallel  position  plane  for  plane. 
On  similar  faces  the  etch  figures  will  be  alike,  and  on  dissimilar 
faces  will  be  unlike. 

*  By  pressure  alone  three  cracks  diagonal  to  these  are  developed. 

f  Growth  and  solution  appear  to  be  reciprocal  and  the  predominating  faces  of 
crystals  growing  in  a  solvent  are  the  very  planes  which  oppose  greatest  resistance  to 
solution  in  that  solvent. 


PERMANENT.  DEFORMATION 


193 


The  figures  produced  by  solutions  of  different  strength  are  not 
necessarily  exactly  the  same,  and  with  different  solvents  there 
may  be  a  still  greater  difference,*  as  strikingly  illustrated  by  the 
tests  of  Baumhauer  f  upon  apatite  with  hydrochloric  and  sul- 
phuric acids.  Fig.  310  represents  the  etch  figures  in  general,  but 


FIG.  310. 


FIG.  311. 


the  basal  planes  treated  with  hydrochloric  acid  show  side  by  side, 
dark  deeply  etched  figures,  usually  consisting  of  a  negative  third 
order  pyramid  truncated  by  the  base  a,  Fig.  311  ;  and  lighter  less 
deep  figures,  usually  a  positive  third  order  pyramid,  ft,  Fig.  311. 

With  100  per  cent,  acid  the  dark  a  figures  are  negative,  but  ap- 
proximately second  order  pyramid,  while  the  lighter  images  ft  are 
positive  and  also  near  the  second  order.  If  weaker  solutions  are 
used,  the  figures  are  differently  oriented.  With  sulphuric  acid 
still  different  orientations  are  obtained. 

The  following  are  the  average  values  obtained  for  e,  Fig.  31 1,  for 
different  strengths,  100%  HC1  being  Sp.  Gr.  1. 1 30  and  100%  H2SO4 
aSp.  Gr.  of  1.836. 

ioo                   60                   50                   20                   10                     5  i 

per  cent.  per  cent.        per  cent.  per  cent.  per  cent.        per  cent.  per  cent. 

a  figures  HC1      (— )  27°2o'  (— )  22°57'     (— )  2o°48'  (— )  l8°44'  (— )  i8°2i'  (— )  i8°5'  (— )  I7°34' 

j3  figures  HCl     (+)27°io'  (-f)28°s6'    (+)26°4o'  (  +  )28°3i'  (+)28°3i'  (— )27°4i'  (— )  i7o34' 

figures  HSS04      (+)i3°7'  (+)    8°35'    (+)itoS°  (-)  i9°39'  (-)  l6°S*' (-) '3°  6  ' 

SYMMETRY  OF  ETCH  FIGURES. 

Whatever  the  solvent,  the  etch  figures  conform  to  the  symmetry 
of  the  class  to  which  the  crystal  belongs,  and  are  rarely  the  limit 

*  For  this  reason  it  is  not  safe  to  assume  any  relation  between  the  shapes  of  the 
etch  figures  and  of  the  crystal  molecule. 
f  Ber.  Ak.  Milnchen,  1887  ,  p.  457. 


194 


CHARACTERS  OF  CRYSTALS. 


forms  common  to  several  classes.  Although  the  first  developed  etch 
figures  may  be  the  simple  forms  bounded  by  the  planes  of  greatest 
resistance  to  solution,  these  are  soon  modified  by  other  planes 
with  complex  indices,  because  the  saturated  solution  in  each  cavity 


FIG.  312. 


FIG.  313. 


is  replaced  much  more  slowly  near  the  bottom  than  near  the 
border,  and  the  attack  is  slower.  The  etch  figures  serve,  there- 
fore, as  an  important  means  (perhaps  next  to  geometric  form  the 
most  important)  for  determining  the  true  grade  of  symmetry  of  a 
crystal.  For  instance,  Figs.  312  and  3 1 3  show  the  shape  and  direc- 
tion of  the  etch  figures  on  cubes  of  galenite  and  sylvite,  respec- 
tively. Fig.  314  shows  the  planes  and  axes  proper  to  class  32,  the 
highest  grade  of  symmetry  in  the  isometric  system.  Evidently  the 
etch  figures  of  galena  satisfy  all  of  these,  whereas  those  of  sylvite 
are  not  symmetrical  to  the  planes  of  symmetry,  but  are  to  the  axes. 
That  is,  galenite  belongs  to  class  32,  sylvite  to  class  29. 


FIG.  314. 


FIG.  315. 


Similarly  Figs.  316  and  317  show  the  shape  and  direction  of  etch 
figures  on  rhombohedra  of  calcite  and  dolomite,  respectively.  Fig. 
315  shows  the  planes  and  axes  of  symmetry  of  class  21.  The  etch 
figures  of  calcite  satisfy  all  of  these,  while  those  of  dolomite  are 


PERMANENT  DEFORMATION. 


195 


not  symmetrical  to  the  planes  of  symmetry,  but  are  to  the  axes. 
That  is,  calcite  belongs  to  class  21,  dolomite  to  class  17. 


FIG.  316. 


FIG  317. 


Similarly,  the  etch  figures  of  wulfenite  Fig.  318  show  the  min- 
eral to  belong  to  class  10,  and  the  etch  figures  of  pyroxene  Fig. 
319  show  it  to  belong  to  class  5. 
MANIPULATION. 

No  general  rule  can  be  given,  it  being  principally  a  question  of 
ease  of  solution.  The  operation  may  consist  merely  in  a  slight 
pressure  from  a  rag  moistened  with  water,  or  the  sliding  of  the 
crystal  across  a  moistened  spot  in  smooth  filter  paper,  or,  more 
frequently,  the  crystal  is  immersed  for  various  periods  in  one  of 
the  mineral  acids,  caustic  alkalis,  or  the  mother  liquor,  hot,  warm 
or  cold,  dilute  or  concentrated.  High  pressure  steam,  water,  hydro- 
fluoric acid,  a  pasty  solution  of  caustic  potash  at  100°  to  150°  C., 
or  even  a  red-hot  fusion  of  acid  potassium  sulphate  and  fluorite 
have  been  used. 


FIG  318. 


FIG.  319. 


Natural  faces  are  usually  most  satisfactory  for  etching ;  cleavages 
are  sometimes  successfully  etched.  The  etch  figures  are  usually 
microscopic,  and  if  large  are  not  apt  to  be  sharp  and  distinct. 


196 


CHARACTERS  OF  CRYSTALS. 


CORROSION  FACES. 

The  continued  action  of  a  solvent  may  dissolve  away  certain  crys- 
tal edges,  replacing  them  by  planes  conforming  in  symmetry  to 
that  of  the  crystal.  These  planes  have  been  called  corrosion  faces. 


HARDNESS. 

The  resistance  of  a  smooth  plane  surface  to  abrasion  is  called 
the  hardness,  and  is  commonly  recorded  in  terms  of  a  decimal 
scale  of  ten  common  minerals  selected  by  Mohs.  In  the  light  of 
the  later  more  exact  methods  it  is  seen  that  there  is  no  even  ap- 
proximately common  difference  in  hardness  between  neighboring 
members*  of  this  scale. 

More  exact  methods,  in  which  greater  uniformity  in  pressure, 
cutting  edge,  condition  and  position  of  surface,  etc.,  have  been 
used  by  various  experimenters.  Exner  f  and  others  supported  the 
crystal  upon  a  little  carriage  moving  on  a  horizontal  track.  The 
face  to  be  tested  was  horizontal,  the  pressure  p  on  the  scratching 
point  of  steel  or  diamond  was  vertical  and  the  carriage  was  moved 
back  and  forth  by  a  weight  W.  The  method  preferred  by  Exner 
was  with  ^constant  to  determine  the  pressure/  necessary  to  pro- 
duce a  visible  scratch. 

From  the  results  obtained  Exner  constructed  the  so-called 
hardness  curves,  for  instance,  Fig.  320  represents  a  dodecahedral 
cleavage  face  of  sphalerite,  radii  are  laid  off  from  the  center,  each 
of  a  length  proportionate  to  the  value  of  /,  for  that  direction  and 
by  connecting  their  ends  a  symmetrical  figure  is  obtained,  show- 
ing six  directions  of  maximum  and  six  of  minimum  hardness. 

*  Jaggar  gives  the  following  comparison  of  the  hardness  of  the  minerals  of  the  Mohs 
scale  Amer.  Journ.  Sci.,  IV.,  411,  1897 : 

Scale  of                       Pfaff  by  Jaggar  by  Rosiwal  by 

Mohs.                      boring  with  a  boring  with  a  grinding  with   a 

diamond  point.  diamond  point.  standard  powder. 

1.  Talc,  laminated 

2.  Gypsum,  crystallized 12.03  .04  .34 

3.  Calcite,  transparent 15.3  .26  2.68 

4.  Fluorite,  crystalline 37.3  .75  4-7° 

5.  Apatite,  transparent 53.5  1.23  6.20 

6.  Orthoclase,  white  cleavable 191.  25.  28.7 

7.  Quartz,  transparent 254.  40.  149. 

8.  Topaz,  transparent 459.  152.  138. 

9.  Sapphire,  cleavable 1000.  1000.  1000. 

-j-  Harte  an  Krystallflachen,  38,  60,  103,  164. 


PERM  AN  EN  T  DEFORM  A  TION. 


197 


Similarly  in  Fig.  321  a  basal  plane  of  barite  shows  a  figure  of 
lower  symmetry. 

The  results  of  Exner  show  that  the  variations  in  hardness  ob- 
served in  any  crystal  are  dependent  upon  the  cleavages.  Faces 
not  cut  by  cleavage  planes  have  constant  hardness  in  all  directions, 
while  faces  intersected  by  cleavage  planes  show  minimum  hard- 
ness parallel  to  the  intersection  with  the  cleavage  plane  and  if  the 
cleavages  are  of  unequal  ease  the  minimum  hardness  is  parallel  to 
the  plane  of  easiest  cleavage.  So  absolute  is  this  relation  that  it 
may  be  expressed  algebraically. 

Very  similar  apparatus  was  used  by  Franz  and  Turner,*  the 
mineral,  however,  being  fixed  and  the  point  moving. 


FIG.  320. 


FIG.  321. 


PfafTf  drew  a  diamond  splinter  of  definite  shape  100  times  back 
and  forth  in  one  place,  then  shifted  and  repeated,  widening  the 
groove.  The  loss  in  weight  of  the  crystal  for  the  same  number  of 
movements  of  the  diamond  over  the  same  area  and  with  a  constant 
pressure  serve  as  approximate  values  for  hardness,  i.  e.,  hardness  is 
inversely  as  the  loss  in  weight.  PfafT  also  used  a  revolving  diamond 
point.  For  equally  deep  penetration  the  hardness  was  as  the  num- 
ber of  revolutions. 

Jaggar  J  designed  an  attachment  to  the  microscope,  in  which 
the  point  of  a  cleavage  tetrahedron  of  diamond  is  rotated  at  a  uni- 
form rate  and  under  uniform  pressure  until  a  cut  of  uniform  depth 
is  obtained  (measured  by  focusing  on  the  rulings  of  a  Zeiss  mi- 
crometer glass,  which  is  slightly  inclined  and  follows  the  downward 
movement  of  the  diamond  point).  The  number  of  rotations  of 
the  point  varies  as  the  resistance  of  the  mineral  to  abrasion  by 
diamond. 

*  Proc.  Phil.  Soc.  Birmingham,  1886. 
\  Ber.  Ak.  Munchen,  1883,  1884. 

\  Microsclerometer,  for  determining  Hardness  of  Minerals. — T.  A.  Jaggar,  Jr. — 
Amer.  Jour.  •»«'.,  IV.,  Vol.  IV.,  399,  1897. 


198  CHARACTERS  OF  CRYSTALS. 

The  effect  of  grinding  has  also  been  used  as  a  test  for  hardness, 
for  instance,  Jannetaz  and  Goldberg*  employed  a  so-called  "  Us- 
ometer,"  consisting  of  a  rotating  grinding  disc,  upon  which  four 
plates,  the  hardness  of  which  is  to  be  determined,  were  pressed  by 
normally  acting  weights,  the  loss  of  weight  of  each  giving  the  rela- 
tive hardness. 

THE  METHODS  OF  STATIC  PRESSURE. 

Following  the  definition  of  hardness  given  by  Hertz,  Auerbach 
devised  a  method  in  which  a  piano  convex  lens  of  any  mineral  is 
pressed  against  a  horizontal  plate  of  the  same  mineral.  Both  are 
bent  and  touch  throughout  a  circular  space,  and  for  some  pressure 
P  the  elastic  limit  is  reached,  at  which  there  is  produced,  if  the 
mineral  is  brittle,  a  circular  fissure,  or,  if  the  mineral  is  tough,  a 
circular  permanent  indentation. 

According  to  Auerbach  f  the  limit  pressure  Pl  upon  a  square 
mm.  of  surface  varies  with  the  radius  p  of  the  lens,  but  the  product 
of  Pl  into  the  cube  root  of  this  radius  p  is  essentially  constant  and 
may  be  called  the  Absolute  Hardness,  that  is 


On  this  basis  he  determines  J  the  absolute  values  of  the  Moh's 
scale  to  be:  corundum,  1,150;  topaz,  525  ;  quartz,  308;  orthoclase, 
253;  apatite,  237;  fluorite,  no;  calcite,  92 ;  gypsum,  14. 

*  Ass.  franc,  p.  /.  avanc.  d.  sc.,  IX.,  Aug.,  1895. 

fF.  Auerbach.—  Wied.  Ann.,  XLIIL,  61 ;  XLV.,  262,  277. 

I  Wied.  Ann.,  LVIIL,  357. 


APPENDIX. 

SUGGESTED   OUTLINE  OF  A   COURSE  IN    PHYSICAL   CRYSTALLOGRPHY. 
PRELIMINARY  EXPERIMENTS. 

In  order  to  secure  systematic  work  the  following  outline  of  experiments 
has  been  prepared : 

A.   GEOMETRICAL   CHARACTERS. 

Preliminary  practice  should  be  given  with  crystal  models  in  the  study 
of  the  thirty  two  classes  and  the  Miller  indices,  also  in  the  use  of  the 
zonal  equations  and  in  stereographic  projections.  The  hand  goniometer 
and  the  743  model  set  of  Krantz  furnish  excellent  practice. 

1.  MEASUREMENT  OF  CRYSTALS.     (Pp.  62-75.) 

Practice  should  first  be  given  in  accurate  adjustment  and  measurement 
of  a  known  angle,  for  instance  the  cleavage  angle  of  calcite. 

Crystals  of  known  system  may  then  be  measured,  the  elementary  planes 
(p.  13)  being  chosen  and,  by  a  preliminary  examination,  the  zones  noted 
and  the  angles  selected  which  it  will  be  necessary  to  measure  in  order  to 
determine  the  elements  of  the  crystals  and  the  indices  of  the  faces. 

Practice  should,  if  possible,  be  given  : 
NJ  (a)  With  horizontal  circle  goniometer.     Babinet  or  Fuess. 

(b)  With  vertical  circle  goniometer.     Wollaston,  Mallard  or  Groth. 

*(^)  With  two  circle  (theodolite)  goniometer. 

2.  DETERMINATION  OF  ELEMENTS,  in  part  by  spherical  triangles  (p.  6),  in 
part  by  special  formulas.    (Pp.  13-15,  34,  38,  45,  56.) 

3.  DETERMINATION  OF  INDICES  AND  ANGLES.     In  so  far  as  possible  by 
zonal  equations  (pp.  1 7-20)  but  also  by  spherical  triangles  and  spe- 
cial equations. 

4.  CRYSTALS  PROJECTION  OR  DRAWING. 

In  addition  to  the  free  -hand  perspective  and  stereographic  projections 
needed  in  the  measuremeut,  accurate  drawings  by  the  various  methods 
should  be  prepared,  especially  : 

(a)  Stereographic  projection.     (Pp.  20-24.) 

(b)  Clinographic  prospective.     (Pp.  76-84.) 

*  Chas.  Palache.     Am.  Jour.  Sci.     S.  4,  Vol.  II.,  p.  279,  1896. 


200  CHARACTERS  OF  CRYSTALS. 

B.    OPTICAL  CHARACTERS. 
>/5.  DOUBLE  REFRACTION  AND  POLARIZATION.     (Pp.  97.100.) 

With  mounted  calcite  rhombs.     (P.  97.) 

(#)  A  ray  of  common  light  through  one  rhomb,  is  split  into  two  rays  of 
approximately  equal  intensity  which  remain  always  in  a  principal  section. 

(b)  If  one  of  these  rays  is  sent  through  a  second  calcite  rhomb,  the 
two  resultant  rays  vary  in  intensity  as  if  due  to  a  resolution  of  a  vibration 
parallel  (or  perpendicular)  to  the  principal  section  of  the  first  rhomb, 
into  components  parallel  and  perpendicular  to  the  principal  section  of  the 
second  rhomb. 

V  6.  DETERMINATION   OF   INDICES   OF   REFRACTION,  with  monochromatic 

light. 

Graphic  determination  (pp.  86-90)  of  the  direction  of  the  refracted 
ray  and  of  the  ray  incident  at  the  critical  angle  should  be  made  with  as- 
sumed indices,  after  which  measurements  should  be  made  both  upon 
singly  and  doubly  refracting  crystals  by  : 

(a)  Prism  method.     (Pp.  88-90,  103,  146-147.) 

(£)  Total  reflection  methods.     (Pp.  90-95,  104,  i47d.) 

(c*)  Displacement  method.     (Pp.  120.) 

If  singly  refracting,  the  substance  will  yield  the  same  index  for  all  di- 
rections. If  doubly  refracting  the  principal  indices  will  only  be  yielded 
in  certain  directions. 

v  7.  PROPUCTION  OF  POLARIZED  LIGHT.     (Pp.  105-110.) 

Examination  of  Nicol  and  Hartnack  prisms  (double  refraction  and 
total  reflection),  tourmaline  pincers  (double  refraction  and  absorption), 
glass  plate  polarizers  (reflection  or  refraction).  Polariscopes  (combina- 
tions of  two  polarizers),  including  older  types  Norremberg,  and  later 
types  of  Universal  Apparatus,  and  Polarizing  Microscopes. 

N  8.  DETERMINATION  OF  EXTINCTION  (VIBRATION)  DIRECTIONS.     (Pp.  117, 

J450 

With  polariscope  (or  polarizing  microscope)  and  monochromatic  par- 
allel light,  sections  between  crossed  nicols  are  black  when,  Fig.  237, 
tp  =  90°,  180°,  270°,  360°,  and  midway  between  show  brightest  illumi- 
nation. 

With  white  light  use  a  test  plate  (gypsum  red)  or  special  eye  piece.  Re- 
sults are  less  accurate  if  there  is  marked  dispersion. 

The  section,  if  too  thick,  will  admit  rays  at  directions  not  normal,  pre- 
venting perfect  extinction. 

Sections  of  enstatite  and  pyroxene  illustrate  respectively  parallel  and  oblique  ex- 
tinction. 


APPENDIX.  201 

9.  INTERFERENCE  OF  PLANE  POLARIZED  RAYS.     (Pp.  110-115.) 

(a)  With  monochromatic  parallel  light  and  crossed  nicols,  extinction 
(throughout  revolution)  takes  place  when    J  =  A,    2/1,  3^,  etc.,  and  for 
J  =  i/l,  |-A,  |^  there  is  the  brightest  illumination. 

(b)  With  white  parallel  light  Newton's  colors  of  i°,   2°,   3°,  etc., 
order  result. 

Wedges  of  quartz,  gypsum,  or  mica,  or  plates  of  different  thickness  of  any  mineral 
serve  for  illustration. 

,/      10.  DETERMINATION  OF  A  AND  n^  —  n.     (Pp.  118-119.) 

By  the  v.  Federow  mica  wedge,  quartz  wedge  or  Babinet  compensator 

J  is  found  and  checked  by  color  chart.     Obtain  nl  —  n  by  formula  J  = 

t(n^  —  ;/) ,  /  being  measured. 
Use  the  sections  of  tests  8  and  9. 

\/  ii.  DETERMINATION  OF  VIBRATION  DIRECTION  OF  FASTER  RAY.     (Pp. 
118.) 

By  test  plate  of  mica  or  gypsum  or  by  wedge  of  quartz  or  mica. 
Use  the  sections  of  tests  8  and  9. 

12.  OPTICALLY  ISOTROPIC  CRYSTALS. 

Between  crossed  nicols,  with  either  parallel  or  convergent  light  all 
sections  remain  dark  throughout  rotation.  The  index  of  refraction  is  in- 
dependent of  the  direction. 

Use  sections  garnet,  haiiynite,  nosite,  etc. 

13.  OPTICALLY  UNIAXIAL  CRYSTALS. 

Models  of  positive  and  negative  ray  surfaces  and  indicatrices  should  be 
studied.  (Pp.  100-102.) 

(a)  Sections  normal  c  between  crossed  nicols.     In  parallel  light  the  sec- 
tion (if  not  too  thick)  remains  dark  throughout  rotation.     In  convergent 
monochromatic  light  the  arms  of  the  cross  remain  fixed  during  rotation  of 
the   plate,  the  rings  are  wider  apart  in  thinner  sections,  and  with  white 
light  are  colored  in  the  order  of  Newton's  colors.     (Pp.  110-117.) 

(b)  Sections  parallel  c  :  interference  hyperbolae  in  monochromatic  light 
and  approximate  indices  by  displacement.     (Pp.  116.) 

(c)  Sections  oblique  to  c.     (Pp.  no,  112,  116.) 

(d)  Determine  character  of  ray  surface  by  different  methods.     (Pp. 

I2O-I2I.) 

Sections  of  zircon,  quartz,  calcite,  beryl,  and  apatite  form  a  good  series. 

14.  UNIAXIAL   CYSTALS    IN   WHICH   c  is  A   DIRECTION    OF    CIRCULAR 
POLARIZATION.     (Pp.  122-130). 

(a)  Circular  and  elliptical  polarization,  with  Fresnel  rhombs. 
(£)  Sections  normal  c,  with  parallel  light. 


202  CHARACTERS  OF  CRYSTALS. 

Monochromatic,  determine  the  direction  and  amount  of  rotation  of 
analyzer  to  produce  extinction. 

White,  note  the  sequence  of  colors  on  rotation  of  the  analyzer. 

(V)  In  section  normal  c,  with  convergent  light.     (Pp.  127.) 

Monochromatic,  interference  figure,  giving  the  inter-wound  spirals  with 
*^  undulation  plate,  or  with  superposed  right  and  left  sections. 

White  light,  interference  figures  with  colored  center,  changing  on  rota- 
tion of  analyzer  in  the  same  sequence  as  for  parallel  light. 

(d)  Sections  parallel  or  oblique  to  c  essentially  conform  to  13  (£)  and 
(c). 

Use  sections  of  quartz  and  cinnabar.  Compare  thickness  of  quartz  wedges  cut  re- 
spectively parallel  and  normal  to  c,  which  give  the  same  color. 

15.  BIAXIAL  CRYSTALS. 

Models  of  positive  and  negative  ray  surfaces  and  indicatrices  should  be 
studied.  Given  three  principal  indices  of  refraction  for  yellow  light 
construct  the  three  optical  principal  sections.  (Pp.  132-137.) 

(a)  In  sections  normal  to  the  acute  bisectrix. 

1.  In  parallel  light  essentially  as  in  tests  8  and  9.     The  relation  of  the 
vibration    (extinction)  direction   to  any  cleavage  or  crystalline  outline 
should  be  noted.     (Pp.  137,  138.) 

2.  In  convergent  monochromatic  light  leminiscates  depend  upon  thick- 
ness, but  the  distance  apart  of  the  axial  points  does  not.     (Pp.  138,  139.) 

3.  In  convergent  white  light  determine  character  of  dispersion.     (Pp. 
140-144.)     Examine  models  of  dispersion. 

4.  Determine  character  of  Ray  Surface.     (Pp.  154.) 

Sections  of  aragonite,  niter,  cerussite,  barite,  calamine,  topaz,  gypsum,  titanite,  di- 
opside,  orthoclase,  borax,  heulandite,  muscovite  and  phlogopite  illustrate  all  phases. 

(£)  In  sections  normal  to  an  optic  axis  in  parallel  monochromatic  light 
there  is  a  uniform  illumination  of  the  field,  in  convergent  monochromatic 
light  will  be  seen  rings  and  one  dark  arm,  which  rotates  in  opposite  di- 
rection to  the  stage.  (Pp.  137-139.) 

#16.  ORIENTATION  OF  a,  b  and  c,  IN  BIAXIAL  CRYSTALS.    (Pp.  145,  146.) 
By  extinction  directions  and  interference  figures  in  variously  oriented 

pairs  of  parallel  planes  (natural  faces,  cleavages  or  planes  obtained  by 

grinding). 

Use  suite  of  sections  or  crystals  of  one  substance. 

17.  DETERMINATION  OF  ANGLE  BETWEEN  OPTIC  AXES.  (Pp.  148-154.) 
(a)  By  measurement  of  apparent  angle  in  sections  normal  to  the  bisec- 
trices either  by  rotation  about  b,  with  universal  apparatus  or  its  equiva- 
lent, in  air  or  in  a  liquid  of  known  index  (pp.  148-150,  152),  or  by 
measurement  of  the  distance  apart  of  the  emerging  axes  in  a  microscope 
or  pclariscope.  (Pp.  151.) 


APPENDIX.  203 

(&)  By  measurement  with  entire  crystal  in  a  liquid  of  approximately 
or  exactly  the  same  index  as  the  crystal.     (Pp.  151.) 
Use  sections  of  muscovite  and  complete  crystals. 

18.  ABSORPTION  AND  PLEOCHROISM.    .(Pp.  96,  130,  131,  155-159-) 

In  isotropic  crystals  there  is  no  pleochroism,  but  there  may  be  either 
color  (unequal  absorption)  or  no  color  (equal  absorption).  The  absorp- 
tion is  independent  of  the  direction. 

In  uniaxial  crystals  there  is  no  pleochroism  for  transmission  parallel 
c,  but  there  may  be  pleochroism  in  other  directions  with  greatest  differ- 
ences for  transmission  normal  to  c . 

In  biaxial  crystals  the  greatest  differences  in  color  are  for  rays  vibrating 
parallel  to  certain  directions  which  in  orthorhombic  crystals  are  a  b  c,  in 
monoclinic  crystals,  •  one  is  b  and  in  triclinic  they  are  not  related  to  the 
crystal  axes. 

Sections  of  vesuvianite,  tourmaline,  penninite,  iolite,  andalusite,  chrysoberyl,  epi- 
dote,  titanite  and  axinite. 

Absorption  tufts  in  epidote  and  andalusite. 

^19.  MICA  COMBINATIONS.     (Pp.  161-162.) 

Production  of  uniaxial  interference  figure  and  rotation  of  plane  of  vi- 
bration by  piling  mica  plates. 

C.    THERMAL   CHARACTERS. 
20.  CONDUCTIVITY.     (Pp.  164-165.) 

Illustrate  surface  conductivity  by  methods  of  Rontgen  and  Voigt,  obtain- 
ing circular  figures  in  isotropic  or  in  uniaxial  normal  to  c ,  but  in  other 
sections  ellipses  in  which  note  relation  of  axes  of  ellipse  to  crystal  axes. 

*2i.   EXPANSION.     (Pp.  166-171.) 

Indirect  determination  by  observing  the  effect  of  heat  upon  the  optical 
characters,  especially  the  effect  upon  the  position  of  the  optic  axes. 
(P.  170.) 

Sections  of  gypsum. 

Indirect  determination  by  observing  change  in  diedral  angle  with  cal- 
cite  cleavage.  (P.  169.) 

D.   ELECTRICAL  AND  MAGNETIC  CHARACTERS. 
22.  MAGNETIC  INDUCTION.     (Pp.  172,  175.) 

Suspension  between  the  poles  of  an  electromagnet  of  a  small  mass  no 
too  unequal  in  dimensions. 

Isometric   Crystals  indifferent  equilibrium  for  all  positions. 


204  CHARACTERS  OF  CRYSTALS. 

Tertragonal  or  Hexagonal  Crystals. 

With  c  vertical  all  positions  are  in  indifferent  equilibrium,  but  with  c 
horizontal  it  takes  either  axial  or  equatorial  position. 

Note  -j-  and  —  paramagnetic  or  diamagnetic  character. 

Crystals  without  any  axis  of  magnetic  isotropy  (orthorhombic,  mono- 
clinic,  triclinic). 

Determine  position  and  relative  strengths  of  magnetization  axes. 
(P-  1730 

Experiments  illustrating  electrical  transmission  (p.  175),  conductivity 
(p.  175),  thermoelectric  currents  (p.  176),  or  dielectric  induction  (p- 
177),  may  be  devised  when  apparatus  is  available. 


23.  PVROELECTRICITY.       (Pp.    180-182.) 

Qualitatively  by  Kundt's  dusting  with  sulphur  and  minium  upon  crystal 
while  undergoing  uniform  change  of  temperature. 
Quantitatively  by  self -charging  electrometer. 
Crystals  of  tourmaline  and  quartz. 

24.  PIEZOELECTRICITY.     (Pp.  182-183.) 

Qualitatively  by  Kundt's  method,  developing  the  strain  either   by  un- 
equal heating,  or  by  direct  pressure. 
Basal  Sections  of  Quartz. 

E.    CHARACTERS  DEPENDENT  UPON  ELECTICITY  AND  COHESION. 

27.  ELASTIC   DEFORMATION   FROM  PRESSURE  IN  ONE  DIRECTION.     (Pp. 
185-188.) 

Effect  of  pressure  in  one  direction  upon  the  optical  characters  of  cubes 
of  glass  and  of  different  minerals. 

28.  CLEAVAGE  ANG  GLIDING  PLANES.     (Pp.  189-191.) 
In  Calcite,  Pyroxene,  Stibnite  and  the  Micas. 

29.  ETCH  FIGURES     (192—196.) 

Examination  of  suite  of  Apatite,  Calcite,  Calamine,  etc. 


SYSTEMATIC  EXAMINATION*  OF    THE  CRYSTALS  OF  ANY 

SUBSTANCE. 

The  objects  are  to  obtain  a  record  of  the  physical  characters  for  differ- 
ent directions,  and  to  determine,  from  the  physically  equivalent  direc- 
tions, the  grade  of  structural  symmetry. 

The  best  crystals  (p.  69),  are  chosen,  the  symmetry  judged  by  exami- 
nation with  a  hand  glass  or  microscope  and  checked  by  an  approximate 

*  Based  upon  Groth,  Phys.  Kryst.,  III.,  537-543. 


APPENDIX.  205 

determination  of  extinction  directions  (p.  117,  145),  and  possibly  of  in- 
terference figures  (115,  127,  138).  Great  assistance  may  here  be  given 
by  a  rotation  apparatus  (p.  151,  152). 

The  crystals  are  sketched  and  letters  assigned  to  the  faces.  Elemen- 
tary planes  (p.  13)  are  chosen  parallel' to  cleavages  (p.  189)  or  according 
to  prominence. 

GEOMETRIC  CHARACTERS  OR  SYMMETRY  OF  GROWTH. 

The  crystals  are  mounted,  centered  and  measured  (pp.  63-70),  by 
zones,  precedence  being  given  to  the  angles  between  the  elementary 
planes.  A  record  is  kept,  as  described  (p.  71),  and  also  upon  a  free 
hand  stereographic  projection  (p.  20).  A  quality  mark  is  assigned 
(p.  70),  to  each  reflection  and  used  in  the  averaging. 

The  elements  are  calculated  (pp.  30,  34,  38,  45,  56,  61)  by  the  for- 
mulae given  or  by  spherical  triangles  (p.  6).  The  indices  are  determined 
in  general  by  intersecting  zones  (p.  17-19),  and  all  angles  are  calcula- 
ted from  the  indices  and  elementary  angles  by  solution  of  the  spherical 
triangles  (p.  6),  in  the  stereographic  projection,  or  by  special  formulae,  or 
by  zonal  equations,  and  compared  with  the  measured  angles. 

One  or  more  perspective  drawings  are  then  made.     (Pp.  76-84.) 

ETCH  FIGURES,  OR  THE  SYMMETRY  OF  SOLUTION. 

Experience  shows  that  the  symmetry  of  the  etch  figure  is  almost  in- 
variably the  true  structural  symmetry  of  the  crystal.  More  than  one  sol- 
vent should  be  tried  and  the  conditions  varied. 

OPTICAL  CHARACTERS  AND  SYMMETRY. 

For  transparent  crystals  the  optical  characters  are  the  safest  proof  of 
symmetry,  often  proving  apparently  simple  crystals  to  be  complex. 

The  optical  constants  for  transmission  are  a,  ft  and  y  (133)  for  light  of 
different  wave-length  (89)  and  their  orientation  (145)  and  direct  meas- 
urement by  the  prism  method  (88,  103,  146)  or  by  total  reflection  (90, 
104,  147)  for  light  of  each  of  Frauenhofer's  lines  is  sometimes  made. 

More  frequently  the  determination  is  limited  to  the  orientation  and  to 
constants  dependent  on  a,  ft  and  ^,  such  as  retardation  (118),  strength  of 
double  refraction  (ii9),character  of  ray  surface  (120,  154),  and,  if  biaxial, 
angle  between  optic  axes  (148).  These  determinations  are  for  sodium 
flame  and  possibly  for  lithium  and  thallium  (89).  If  biaxial  the  interfer- 
ence figure  should  also  be  observed  in  white  light  for  dispersion  (132-144). 

Absorption  and  pleochroism  are  determined  (96,  130,  155). 

If  the  crystals  show  a  rotation  of  the  plane  of  polarization  (123)  the 
amount  (129)  and  direction  (128)  is  determined. 


206  CHARACTERS  OF  CRYSTALS. 

THE  REMAINING  PHYSICAL  CHARACTERS. 

The  most  frequently  used  tests  remaining  are  the  effects  of  heat,  elec- 
tricity and  pressure  upon  the  optical  characters.  In  addition,  however, 
and  especially  in  minerals  opaque  to  light  the  following  may  be  determined 
without  very  elaborate  apparatus. 

(#)  The  heat  conductivity,  Rontgen  or  Voigt  methods  (p.  164). 

(b)  The  pyroelectric  charges,  Kundt's  method  (p.  180). 

(<:)  Cleavage  (p.  189),  gliding  planes  (p.  190),  percussion  figures 
(p.  192). 

Thermal  expansion,  Magnetic  Induction,  Electrical  conductivity, 
Thermoelectric  currents,  Dielectric  induction,  Elastic  deformation  rarely 
form  part  of  an  investigation. 

The  ascertained  facts  may  be  conveniently  recorded  as  suggested  by 
Groth*  and  constitute  a  Crystallographic  description. 

(#)  Class  of  Symmetry. 

(£)  Geometric  Characters. 

Elements,  Enumeration  of  forms  and  description  of  habit  with  per- 
spective drawings,  Twinnings,  Conditions  of  temperature  solution,  etc., 
in  production  of  crystals. 

(V)  Cleavages,  Tabulation  of  measured  and  calculated  angles,  Gliding 
Planes  and  Etch  Figures. 

(d}  Optical  Characters. 

(i)  Other  physical  characters. 

*  Phys.  Kryst,  III.,  543. 


INDEX. 


Abbe,  total  reflectometer,  95. 

Absolute  hardness,  198. 

Absorption,  96,  130,   155. 

Absorption  bands,  1 60. 

Absorption  axes,  155. 

Absorption,  measurement'of   relative,  156. 

Absorption  tufts,  148. 

Adjustments,  Fuess'  goniometer,  68. 

Airy's  spirals,  128. 

Amplitude,  85. 

Analyzer,  107. 

Analogue  pole,  180. 

Angle  alteration  by  expansion,  1 68. 

Angles  between  crystal  axes,  determining, 

*3- 

Angles  between  axial  planes,  13. 
Angles  between  optic  axes,  148. 
Angles  between  any  two  planes,  46. 
Angles,  constancy  of  interfacial,  3. 
Angles  and  indices,  relations  between,  29, 

39,  46,  62. 
Antilogue  pole,  1 80. 
Auerbach,  absolute  hardness,  198. 
Apparent  axial  angle,  148. 
Axes,  8,  ii. 
Axes,  changing,  19. 
Axes,  equivalent,  II. 
Axes  of  elasticity,  101. 
Axes,  optic,  97,  122,  124,  131,  133,  148. 
Axial  angle,  measurement,  148. 
Axial  angle,  calculation,  153. 
Axial  angle  changed  by  heat,  170. 
Axial  angle  apparatus,  149. 
Axial  cross,  80. 
Axial  image,  115,  127,  140. 
Axial  plane,  135. 

Babinet,  compensator,  118. 

Basal  pinacoid,  32,  37,  44,  50,  54. 

Basal  plane,  32,  37,  44,  50,  54. 

Basal  section,  superposition,  121. 

Becke,  method  indices  of  refraction,  120. 

Becquerel,  magnetism,  172. 

Beijerinck,  electrical  conductivity,  176. 

Bending  rods  of  crystals,  186. 

Bergman,  Torbern,  3. 

Bernhardi,  7. 

Bertrand,  eyepiece,  117. 

Bertrand,  lens,  150. 

Bertrand,  prism,  105. 

Biaxial  crystals,  132. 

Biot,  quartz  plate,  130. 

Bipyramids,  44,  50,  54. 

Bipyramidal  classes,  36,  41,  42,  48,    49, 

.52,  53- 
Biquartz,  130. 


Biradials,  133. 
Bisectrices,  136,  148. 
Bisphenoids,  44. 
Bisphenoidal  classes,  35,  40. 
Body  colors,  160. 

Bolzman,  method  dielectricity,  179. 
Bose,  electrical  polariscope,  175. 
Brightest  illumination,  73. 
Bromnaphthalin,  a,  95. 

Calculation  of  crystals,  28,  33,  38,  45,  55, 

6l. 

Centering,  65,  71. 
Character  of  ray  surface,  120,  154. 
Circular  polarization,  95,  122,  126. 
Cleavage,  5,  188. 

Clinographic  parallel  perspective,  79. 
Collimator,  65. 
Cohesion,  188. 

Color  distribution  in  lemniscates,  140,  143. 
Color  rings,  115. 
Color  tints,  96. 
Colors,  interference,  113. 
Colors,  surface  and  body,   160. 
Compensators,  118. 
Conical  refraction,  135,  136. 
Constants,  optical,  103,  145. 
Composite  crystals,  73. 
Contact  goniometer,   3,  63. 
j  Conductivity,  electrical,  175. 
Conductivity,  thermal,  164. 
Convergent  light,  107. 
Corrosion  faces,  196. 
Critical  angle,  91. 
Cross  and  rings,   115. 
Crossed  dispersion,  143. 
Crystal,  definition,  I. 
Crystals  in  rock  sections,   154. 
Crystal  carrier,   67. 
Curie,  piezoelectric  method,   182. 

Decretion,  7. 

Deformation,    elastic,   185. 

Deformation,  permanent,   188. 

Deformation,  effect  on  optical    characters 
187. 

Derivation,  II. 

Diamagnetism,   172. 

Dichroscope,  131. 

Dielectric  coefficients,   179. 

Dielectric  induction,   177. 

Dihexagonal  bipyramid,  class  of,  53. 

Dihexagonal  prism,  50,  54. 

Dihexagonal  pyramid,  class  of,  53. 
|  Diploid,  class  of,   58. 
I  Dispersion  of  axes,  132,  140. 


208 


INDEX. 


Dispersion  of  the  bisectrices,  132,  141,  144, 
Displacement  method,   120. 
Ditetragonal  bipyramid,  class  of,  42. 
Ditetragonal  prism,  44. 
Ditetragonal  pyramid,   class,    44. 
Ditrigonal  bipyramid,  class,  49. 
Ditrigonal  prism,  50. 
Ditrigonal  pyramid,  class,  48. 
Ditrigonal  scalenohedron,  class,  48. 
Dogtooth  spar,  3. 
Domes,  32,  37. 
Dome  class,  31. 

Double  refraction  in  calcite,  97. 
Double  refraction  by  pressure,  188. 
Double  refraction  of  electric  rays,  175. 
Double  refraction  of  heat  rays,  163. 
Drawing,  76. 
Dull  faces,  73. 

Edges,  82. 

Elastic  deformation  by  pressure,  185. 

Elastic  limit,  188. 

Electric  polariscope,  175. 

Electrical  conductivity,  175. 

Electro-optical  phenomena,  184. 

Elementary  planes,  13. 

Elements  determination,  30, 34, 38, 45, 56,61. 

Elements  of  a  crystal,  13. 

Elliptical  polarization,  122,  123. 

Enantiomorphs,  125,  126. 

Etch  figures,  192. 

Ether,  85. 

Exner's  hardness  curves,  197. 

Expansion  by  electric  charge,  183. 

Expansion  measurements,  167,  168,  169. 

Expansion  by  heat,  166. 

Extension  coefficients,  186. 

Extension  surfaces,  186. 

Extinction,  117,  145. 

Extinction  directions  and  optic  axes,  137 

Fizeau,  expansion  measurements,  167. 
Fluorescence,  160. 
Foucault,  prism,  105. 
Fresnel,  rhomb,  123. 
Fuess,  goniometer,  66. 
Fuess,  microscope,  108. 
Fundamental  form,  9. 
Fundamental  law  of  crystals,  II. 

Gahn,  3. 

Gaugain,  pyroelectricity,  181. 
Glans,  spectrophotometer,  156. 
Gliding  planes,  190. 
Goldschmidt,  projection,  76. 
Goniometer,  application,  3,  63. 
Goniometer,  reflection,  63,  64. 
Goniometers  with  horizontal  axes,  64. 
Goniometers  with  vertical  axes,  66. 
Goniometer,  two-circle,  74. 
Glycerine,  95. 
Graphic  method,  9. 


Grazing  incidence,  91. 
Grinding  new  faces,  104. 
Groth,  goniometer,  66. 
Groth,  universal  apparatus,  149. 
Gulielmini,  Dominico,  3. 
Gypsum  test  plate,  1 1 8. 
Gypsum  red,  first  order,  121. 
Gyroid,  class  of,  57. 

Haidinger,  dichroscope,  131. 

Hankel,  pyroelectric  method,  180. 

Hardness,  196. 

Hardness  absolute,  198. 

Hardness  curves,  197. 

Hartnack,  prism,  105. 

Hausmann,  8. 

Hatty,  Abbe,  5. 

Heat  conductivity,  164. 

Heat  rays,  163. 

Heat  expansion,  166. 

Hemihedrism,  25. 

Hexagonal  bipyramid,  50,  54. 

Hexagonal  bipyramid  third  order,  class 
of,  52. 

Hexagonal  crystals,  projection  and  calcu- 
lation, 55. 

Hexagonal  prism,  50,  54- 

Hexagonal  pyramid,  50?  54- 

Hexagonal  pyramid,  third  order,  class  of 
the,  52. 

Hexagonal  system,  46. 

Hexagonal  trapezohedron,  class  of,  52. 

Hexahedron,  60. 

Hexoctahedron,  class  of,  58. 

Hextetrahedron,  class  of,  58. 

Historical  introduction,  I. 

Horizontal  dispersion,  142. 

Huyghen's  construction,  80,  87. 

Homogeneous  deformation,  185. 

Hyperbolae,  interference, '116. 

Imbedded  crystals,  74. 

Inclined  dispersion,   142. 

Index  of  refraction,  86. 

Index  of  refraction  by  prism  method,   88, 

103,  146. 

Index  of  refraction  by  total  reflection,  90, 

104,  147. 

Indicatrix,  optical,  IOI,  133. 

Indices,  see  index. 

Indices  of  planes,  12. 

Indices  of  zones,  17. 

Indices  and  axes,  equation  between,  29,  39, 

46,  62. 

Induction  dielectric,  177. 
Induction,  magnetic,    172. 
Inner  conical  refraction,  136. 
Integrant  molecules,  6. 
Intensity  of  rays,  112,  157. 
Intercepts,  9,  II. 

Interfacial  angles,  constancy  of,  3. 
Interference,  106,  no,   in. 


INDEX. 


209 


Interference  colors,  112. 
Interference  figures,   115,  127,  138. 
Interference  phenomena  uniaxial   crystals 

no. 
Interference  phenonmena,  biaxial  crystals 

137- 
Isometric  crystals,  projection  and  calcula 

tion,  61. 

Isometric  system,   57. 
I  so  tropic  crystals,  85. 

Jaggar,  apparatus,    154. 
Jamitzer,  W.,  2. 

Kepler,  2. 

Kelvin,  theory  of  pyroelectricity,  l8l. 

Kernel,  7. 

Klein,  lens,  150. 

Klein,  Universal  Apparatus,  151,  155. 

Koch,  measurement  extension,  186. 

Kohlrausch,  total  reflectometer,  91. 

Kundt,  method  pyroelectricity,  180. 

Law  of  Babinet,  156. 

Law  of  rational  indices,  7. 

Law  of  symmetry,  3. 

Lemniscates,  138. 

Least  deviation,  89,  103,  146,  147. 

Liebisch,  total  reflectometer,  93. 

Light  rays,  85. 

Linear  expansion,  167. 

Linear  projections,  76. 

Magnetic  induction,  172. 

Magnetization,  relative,  173. 

Mallard,  goniometer,  65. 

Measurement  of  angles,  63,  70. 

Metallic  lustre,  159. 

Metallic  reflections,  159. 

Methylene  iodide,  95. 

Mica  test  plate,  118,  120. 

Mica  wedge,  v.  Federow,  118. 

Mica  combinations,  161. 

Microscope,  Seibert  n  A,  107. 

Microscope,  Fuess  VI,  108. 

Miller,  W.  H.,  9. 

Minimum  deviation,  89,  103,  146,  147. 

Mitscherlich,  goniometer,  66. 

Mohs,  9. 

Molecular  net  structure,  189. 

Monochromatic  light,  89. 

Monoclinic  crystals,  extinction  directions, 
145-  . 

Monoclinic  crystals,  interference  figures, 
141. 

Monoclinic  crystals,  projection  and  calcu- 
lation, 33. 

Monoclinic  dome,  class  of,  31. 

Monoclinic  sphenoid,  class  of,  31. 

Monoclinic  system,  30. 

Narrow  faces,  72. 

Negative  ray  surface,  IOI,  104,  121,   136. 


Negative  crystals,  101,  104,  121,  136. 
Neumann,  F.  C.,  9. 
Nicol,  prism,  105. 

Norremberg,  mica  combination,  161. 
Norremberg,  polariscope,  109. 
Octahedron,  60. 
Optic  axes,  97,  133. 
Optic  axes,  angle  between,  148. 
!  Optical  characters  during  pressure,  187. 
Optical  characters  during  heating,  170. 
Optical  characters  in  electrostatic  field,  184. 
Ordinary  ray,  98. 
Orientation  of  principal  vibration  directions, 

145- 

Orthographic  parallel  perspective,  77. 

Orthorhombic  crystals,  extinction,  140. 

Orthorhombic  crystals,  projection  and  cal- 
culation, 38. 

Orthorhombic  system,  35. 

Outer  conical  refraction,  135. 

Parallel  polarized  light,  106. 

Paramagnetism,  172. 

Parameters,  12. 

Parameters,  changing,  20. 

Parameter  ratios,  14. 

Parting,  191. 

Pentagonal  dodecahedron,  60. 

Percussion  figures,  191. 

Permanent  deformation,  1 88. 

Permutations  of  letter  and  sign,  36,  42,  54. 

Phosphorescence,  161. 

Piezoelectricity,  182. 

Pinacoids,  fig.  79,  27,  32,  37. 

Pinacoid,  class  of,  36. 

Plane,  or  pedion,  27,  32. 

Plane-parallel  plates,  87. 

Plane  polarized  light,  105. 

Plane  of  polarization,  100. 

Plane  of  vibration,  98. 

Pleochroic  images,  157. 

Pleochroism,  130,  155. 

Pliicker,  magnetization,  173. 

Polariscopes,  106. 

Polarizers,  105. 

Polarization  apparatus,  1 06. 

Polarization  colors,  112. 

Polarization  of  electric  rays,  175. 

Polarization  of  light  in  calcite,  98. 

Polarization  of  heat  rays,  163. 

Polarized  light,  105,  122. 

Polarizing  microscope,  107,  loS. 

Pole  of  a  face,  16. 

Pole,  position  of  any,   29,  34,  39,  45,  56, 

62. 

Poles,  arc  between,  30,  35,  39,  56. 
Positive  crystals,  IOI,  104,  121,  136. 
Positive  ray  surface,  IOI,  104,  121,^136. 
Pressure  figures,  192. 
Pressure,  uniform,  185. 
Pressure  in  one  direction,  185 
Pressure,  effect  on  optical  characters,  187. 


210 


INDEX. 


Primitive  form,  4,  5. 

Primitive  circle,  20. 

Principal  indices  of  refraction,  103,  132. 

Principal  optical  sections,  132. 

Principal  vibration  directions,  132. 

Prism,  32,  37,  44,  50,  54. 

Prism  methods  indices  refraction,  88,  103, 

146. 

Prismatic  class,  3 1 . 

Projections  in  parallel  perspective,  77. 
Projections,  stereographic,   9,   20,  28,  33, 

38,  45,  55,  61. 

Pulfrich,  total  reflectometer,  94. 
Pyramids,  44,  50,  54. 
Pyramidal    classes,  35,  40,  44,  47,  48,  52. 
Pyroelectricity,  180. 

Quadrant  in  drawing  axial  cross,  84. 

Quality  mark,  70. 

Quarter   undulation  mica  plate,   118,  120. 

Quartz  wedge,  118. 

Quenstedt,  linear  projection,  76. 

Rationality  of  parameters,  7. 

Ray  axes,  133. 

Ray  front,  85. 

Ray  surfaces,  85,  86,  95,  100,  125,  133. 

Ray  surfaces,  character  of,  1 20,  154. 

Reflection  goniometer,  63,  64. 

Refraction  in  biaxial  crystals,  136. 

Refraction  for  normal  incidence,  87. 

Refracting  liquid,  95,  153. 

Retardation,  118. 

Reusck,  mica  combination,  161. 

Rhombic  bipyramid,  class  of,  36. 

Rhombic  bisphenoid,  class  of,  35. 

Rhombic  dodecahedron,  60. 

Rhombic  pyramid,  class  of,  35. 

Rhombohedral  crystals,  47. 

Rhombohedral  class,  47. 

Rhombohedron,  50. 

Rontgen,  conductivity,   164. 

Rohrbach,  solution,  95. 

Rome  Delisle,  3. 

Root,  dielectric  induction,  177. 

Rotation  of  polarization  plane,  123, 128, 129. 

Rotation  by  Reusch  mica  combination,  161. 

Scalenohedral  class,  41,  48. 

Sclerometric  tests,  196. 

Secondary  forms,  4,  5. 

Senarmont,  electrical  conductivity,  175. 

Senarmont,  thermal  conductivity,  164. 

Sensitive  tint,  129. 

Signals,  68. 

Snellius',  construction,  87,  90. 

Soleil,  quartz  plate,  130. 

Sorby,  displacement  method,  1 20. 

Spectroscope  in  index  measuring,  92. 

Spectrum  photometer,  156. 

Sphenoid,  31. 

Sphenoidal  class,  31. 


Spherical  projection,  6. 

Spherical  trigonometry  formulae,  28. 

Static  pressure  tests,  198. 

Steno,  N.,  2. 

Stereographic  projection,  9,  20. 

Strength  of  double  refraction,  119. 

Striated  faces,  72. 

Structure,  7,  162. 

Surface  colors,  159. 

Symmetry  axes,  10. 

Symmetry,  composite,  II. 

Symmetry  grade,  IO,  25. 

Symmetry  of  etch  figures. 

Symmetry  plane,  lo. 

Symbol,  II,  17. 

Symbols,  see  indices  and  index. 

Systems,  9. 

Tangent  principle,  39,  56. 

Tetartoid  class,  57. 

Tetragonal  bipyramid,  44. 

Tetragonal  bipyramid  3°  order,  class,  41. 

Tetragonal  bisphenoid,  44. 

Tetragonal  bisphenoid  3°  order,  class,  40. 

Tetragonal  crystals,  projection  and  calcu- 
lation, 45. 

Tetragonal  prism,  44. 

Tetragonal  pyramid,  44. 

Tetragonal  pyramid  3°  order,  class,  40. 

Tetragonal  system,  40. 

Tetragonal  trisoctahedron,  60. 

Tetragonal  tristetrahedron,  60. 

Tetrahedron,  60. 

Tetrahexahedron,  60. 

Theodolite  goniometers,  74. 

Thermal  characters,   163. 

Thermal  conductivity,  164. 

Thermal  expansion,  1 66. 

Thermoelectric  currents,  176. 

Thirty-two  classes,  crystals,  25. 

Thickness  of  section,  119. 

Thoulet,  solution,  95. 

Tint  of  passage,  129. 

Total  reflection,  90,  104,  147. 

Total  reflectometers,  91. 

Tourmaline  distribution  of  charge,  182. 

Translation,  191. 

Transparent  crystals,  measuring,  72. 

Traube,  attachment  goniometer,  73. 

Trapezohedron,  47. 

Trapezohedral  class,  41,  47. 

Triclinic  crystals,  projection  and  calcula- 
tion, 28. 

Triclinic  crystals,  extinction,  145. 

Triclinic  system,  26. 

Trigonal  bipyramid,  5°- 

Trigonal  bipyramid,  3°  order,  class,  48. 

Trigonal  prism,  50. 

Trigonal  pyramid,  50. 

Trigonal  pyramid  3°  order,  class,  47. 

Trigonal  trapezohedron,  class,  47. 

Trigonal  trisoctahedron,  60. 


INDEX. 


211 


Trigonal  tristetrahedron,  60. 
True  axial  angle,  153. 
Twin  crystals,  73,  83. 

% 

Unsymmetrical  class,  26. 
Uniaxial  crystals,  97. 
Universal  stage  of  v.  Federow,  155. 
Universal  apparatus,  Groth,   149. 
Universal  rotation  apparatus,   Klein,  151, 
155- 

Vibration,  direction  of  faster  ray,  Il8. 
Vibration  plane,  98,  117,  145. 
Vibration,  principal  directions  of,  132, 146. 
Vibration   of  ordinary   and    extraordinary 

ray,  98. 
Voigt,  measurement  expansion,  167 


Wave-lengths,  85,  89,  113. 

Weiss,  8. 

Whewell,  9. 

Wiedemann,  surface  conductivity,  179. 

Wollaston,  goniometer,  64. 

Zone,  1 6. 

Zone  circle,  1 6. 

Zone  plane,  1 6. 

Zone  axis,  17. 

Zone  control,  17. 

Zone  during  expansion,  168. 

Zone,  fourth  face  in  a,  18. 

Zone  indices  or  symbols,  17. 

Zone  through  one  pinacoid,  19. 

Zone  of  two  pinacoids,  19. 

Zone  projection,  23. 

Zonal  relations,  55. 


USE 

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